After viewing many videos and reading the above guides/tips I am throwing my hat it, because many of the videos I think some people are very skilled or very lucky, and you need a method that works for the average player.
Rd 1, I simply boost to where the red robots begin to spawn, and AR mode head shot them, boost into the statue area and circle around killing all but one red robot. I have the music turned off to hear the viper foot steps and when I smash him I shottie him twice. Choose the location of where he dies, simply because the viper drops an upgrade, and if you make a habit of killing him in the same place you will play always knowing where the upgrade is. I personally like on the upstairs area where you begin the round. When you kill the viper that leaves one red robot, take this time to change weapons or breathe. When I first did this method I would swap the Shottie for the Lock on Laser, but soon thought it may be valuable for the last round. Stand on the right most staircase of where you begin the round, by the rubble and kill the robot.(My loadout Heavy Machine gun 200 rds, normal machine gun about 400 rounds, Shottie at 11-12 ammo, all three nades and emp.
Rd 2. When the orb is 90% formed for the Bogey I throw a hand grenade. When it explodes it does perhaps 1/5 of his life which just makes killing him that much easier. I will AR mode and hvy machine gun him until I am at half the threshold of the AR mode. In this process I switch to EMP nades. In a matter of 10-15 seconds the first wave of flying copters should be in mid range proximity. Throw the EMP nade at them, in the majority of my attempts this would kill all five and stun the bogey. From the top of the stair case I would boost roll to the opposite side of the upper level towards the 2nd wave of copters and throw an EMP grenade, once again if you are lucky this will stun the bogey (you don't need to fire at him while stunned its just added bonus). Now it should be just gold and red bots hiding and shooting in the middle area or statue area with the Bogey flying around. Since you rolled to the side of the 2nd wave of copters, right near you is the yellow wall sand bags, this becomes your protective cover if need be. The robots generally won't swarm you and being higher elevation from them prevents them from getting clean shots. So its you, sand bags and bogey, I would simply AR mode Hvy MG the Bogey until he dies, if he gets close lick your lips and pull out the shottie as it does devastating damage to him. When the bogey is down hide behind the sand bags and just pick off the drones with the assault rifle, leave one red drone. Exchange the shotgun for the disk launcher gun, grab some assualt rifle ammo, I grab mine from the three weapons rack at the place the challenge begins. I then run to the half sand bags on the middlish staircase on the left side of where the challenge begins. See Rd 3 reference point. Kill the last red bot to begin round 3.
Rd 3, now some people will tell you to hide behind a far right corner, use the LFE on the gas tank romanov, which pushes him back into the rocket romanov, you then have to shoot the tank with the assault rifle until it explodes, this may damage a drill romanov. I see multiple videos use this method, and when I try this method many issues occur, namely when you throw a grenade at the small 5 red drones it is hard to gauge the accuracy. Secondly after shooting the LFE gun and moving the romanov you have about 10 seconds to shoot the tank, at long range your accuracy needs to be spot on or otherwise this method fails and you are in a pickle because you have 4 romanovs trying to be your friend. The best method to conquer the 3rd round consistently is a youtube video by Dragenyo about 13:15 mins long, at 2:26 is where you should begin round 3. My loadout beginning this map is Hvy mg at about 130 rounds, 600 rd assualt rifle and disk launcher. I throw the hand grenade to kill the red bots like he does, if one gets away let him chill, as he will be the last to die and give you time to set up. You let the romanov with the gas tank get close to you and AR mode shoot his gas tank. You know you mastered this method that Dragenyo shows because it A) kills, the gas tank romanov b) it kills the purple laser Romanov, and c) heavily damages a drill romanov. When I enter AR mode it generally only takes me to half of it using the heavy MG to make the tank explode, thus giving me room to boost dodge. In the video Dragenyo uses the assault rifle and I think it costs him an extra 25% of his boost to do so. After the explosion I roll diagonal right from the current position and I sweep around to where the EMP grenades are sitting behind the sand bags. While doing this the rocket romanov is spraying his machine gun, hitting on a few instances, and a drill romanov may be chasing you. From this position I continue to boost around to the back of the statue in the center circle, perhaps before reaching the back I will AR mode, switch to the disk launcher and aim for the rocket romanov's head. Dragenyo does his variation which is boosting directly at the rocket romanov, I take the long way because a lucky hit or an elevated damage status and then a small hit can trigger overheat mode. When I am behind the statue I will swing out and shoot the disk in AR mode until the head is cut off of the rocket romanov, if I succeed before taking cover yay for me. Now there is a headless rocket romanov whose only real danger is his two small rockets which are heat seeking. At this point I am avoiding the drill romanovs, but looking for the one that is near death (due to the gas tank explosion). I will hvy mg him. You may say why not slice the head, and the simple answer is that if you kill him you remove a piece from the board. My current state is a headless rocket romanov and a full life drill romanov, now I will try decapitating the drill romanov. Once done the map has two headless romanovs, and I have 30-60 seconds to run around and acquire weapons for the next round. I replace the disk launcher with the LFE gun laying on the ground, dropped by some enemy, and restock the heavy machine gun. I will stand on the stair case of where I began round 3, with LFE equipped. (Load out so far, 200 HvyMG, Assualt, 10 LFE, 1 EMP nade, one regular nade.) Once the Romanovs burn out and die Round 4 begins, which for some is the most challenging because you are swarmed.
Rd 4 Based on where I am standing I will see two orbs forming, the only one I am really concerned with is the rolling bots, I make sure I shoot an LFE round to end them. I will AR mode, to make sure this is done right. My goal is to sweep around to where the EMP grenades are and sand bag area, all the while I am briskly going in that direction and maybe once or twice I will AR mode and LFE the blue bots. After sweeping to the location of the EMP grenades, with a large chunk of boost remaining, I will AR Mode and HvyMG the 1st flying bot, this should bring me to about 25% boost. I then dodge boost to the back corner where there are sand bags and an assualt rifle. While you sweep in this direction, take note of your map, the blue bots when they chase you like to line up. When you see this and are in the sand bag/assualt rife area (reference You tube video Vanquish Tactical Challenge 6 by MoluJpn about 5:10 long (2:24-2:50). What MoluJpn did is what I based my movements off. Around the sand bag area my goal is just to boost dodge, AR mode and LFE the blue bots when they bundle up, it may take two shots to kill the blue bots, so don't mind firing rolling, rolling back into AR mode and firing again. When all the blue bots are destroyed at this point one of the scorpions is getting near. I will boost dodge, switch to HvyMG and move to a spot where I can AR Mode the 2nd flying gunner. Now it is two scorpions and myself. Now I have seen many videos, some cool, where you can LFE the scorpions get them close together and use the rocket launcher to finish them quickly. However, make it dull and HvyMg one of them to death. If you are dodging and moving the only real threat is the 4 purple capsules they release from their chest and head towards you like missiles. AR mode and eliminate those purple missiles when you see them. When it is just one Scorpion and myself I often use the Assault rifle to lower him to near death. Firing the hvy mg should have taken most of the clip to kill the 1st one, any remaining ammo I would use on the 2nd one scorpion. When the last scorpion is near death this is when you want to prepare for the next rd. As you can see throughout the 1st 4 rounds I didn't have use use the rocket launcher, the upgrade dropped by the viper, the 2nd set of EMP nades, or the two Lock on Lasers. At this point my method gives you the flexibility and firepower to deal with the bogeys. The first step is to grab the LFE refill, for me this is in the racks where the tact challenge begins, or may be on the ground depending upon which one you picked up. I would refill the Hvy MG near statue or upper level rack. And the assault rifle is going to get swapped based on what you want to have in hand, the Lock on laser or rocket launcher. Understand that when you deal with the rolling robots in the beginning of the 5th round your goal will be to change the LFE with a 2nd weapons, ie the choice you didn't take. When you are loaded up, HvyMG the last scorpion and be positioned at location shown 3:42 of MoluJpn tact challenge video.
Rd 5 Positioning is key as you can quickly, within 20 seconds, roll, AR mode and eliminate the rolling robots with the LFE gun. The 1st two are easy as you are rolling/dodging and when they do appear AR mode and fire. Once the first two orbs have been dealt with dodge roll and AR mode to the other rolling robots, which should be near the vicinity of where the 2nd orb spawned rolling robots. Once these are dealt with you are exactly where you want to be for this challenge, you, 2 bogeys, and minutes of rolling firing to kill them. At this point it is about tips not a walk through, so below are key points.
* Remember for defensive purposes you have at a minimum 3 EMP grenades, I had 4 (1 + 3 bundle) and 3 cigarettes. Many people in videos get in pickles and I don't see them take advantage of throwing a smoke, this sometimes can buy you 20 seconds of calm.
* the attack that will get you when rolling and dodging, often triggering the overheat, making you do the above is the charge up linear laser blast. If you are behind cover stay there as it will NOT hit you. Many videos I see people roll out of cover to avoid it, this is a mistake. When I did my play through this was the one attack that had me scramble to throw an emp or cigarette.
* Focus on one at a time, some people recommend lowering both at the same time, pure stupidity. Two Bogeys means more shots down range. More shots down range means you are dodging more and your chance of getting hit is greatly increased. Eliminate one of them turns it into 1vs 1, which based in round two you killed him solely with a hvyMG and a grenade, now you have Lock on laser and rocket launchers. I killed the blue one first just by chance, but know that he drops a rocket launcher, with the one you had, plus the upgrade you can get a total of 9 rockets. 9 rockets and one red bogey I think anyone can pull off that victory.
* my preference is to AR mode and fire a Lock on Laser and continue rolling. The key strategy is boost consumption, if you stay in AR mode for a long period of time then you are reducing your total boost and when you get hit you will have less time to navigate to cover. So the weapons that you fire and then move are preferable. Every now and then I would hvy MG, especially when they are close range or talking smack chilling in the air. the lock on laser won't kill them as fast as certain weapons but it guarantees hits, quick firing and this allows you to continue to dodge while maintaining a high boost.
* some videos show you how to wreck two bogeys with EMP, Shottie, Hvy Mg and melee. These are skilled players, however the point you take away is that don't be afraid of CQC. Often times the weapons you can take close will do more damage.
*In round 3 the headless drill romanovs will be like bulls in a china shop, they may destroy a lot of cover, they may not. This will direct the number of "ohh crap hide" locations for this round, however my favorite covers are the in between walls of the stair cases for the upper level, if you learn to figure 8 them or travel from end to end it can often make rolling and attacking easier to implement.
* get proficient with the lock on laser, in round 2 I used to kill the bogey with it, literally rolling AR mode firing, rolling AR mode firing back and forth I would drain him in about 20 seconds. This quick strike method can often kill a bogey that much quicker. I definitely would use this if I have the rounds and I am clear, and he is near death, ie 4/5 =16/20 hits for bogey.
* When there is one bogey left it becomes incredibly easier to dodge, cover, and know where the attacks are coming from, I cannot state it enough, focus on one and kill.
It took me about 20 mins to win this round, so patience is sometimes need. Following this routine I became accustomed to the actions I would take, knowing what weapons I would use, where to set myself up. The goal is to eliminate variables and exploit weakness of the enemy. The 1st 3 rounds I can do consistently, the 4th got easier when I stuck to the same tactic. 5th round beginning a few times didn't go my way, so I mastered how to eliminate them. When its u and the bogeys sometimes they get lucky shots. Your goal should be to master and dominate each wave, so you do the 1st and 2nd wave in your sleep. At times on the 3rd round if you hit the tank romanov he will keel back and block the tanks, in this instance your are screwed, so fine tuning each detail is going to allow you to conquer this challenge. Honestly people say this is hard, but I spent most of my time trying different methods, after following this process over a few times ie 3 hrs playing, fine tuning, controlling where enemies went etc, it required less skill on my part to accomplish this challenge. Someone says play God hard to practice for this, that is a waste, because the challenge reacts like Hard mode, so you might as well train in the arena you are going to fight in.