Here are 3 ways you could do this:
1. Co-op with a second controller
2. Co-op with friends
You will not find any UNSC weapons or ammo during this mission and you will not respawn with a UNSC weapon.1. Co-op with a second controller
The fastest and easiest way to get this achievement is to turn on a second controller and start mission 3 (Forerunner) in co-op and use the second character for most of the mission. If you find your main character is teleported to the second character (such as when you enter the cavern with the Banshees) move your main character into a safe place. When you get to the Ghosts near the end of the level, switch back to your main character to finish the mission. If your second character gets killed you will not revert to the last checkpoint like you would do on solo. That is why it is the fastest way to do this as you will never have to repeat anything if you die. It is the easiest way as you can carry and use whichever two weapons you want with your second character, without having to worry about keeping a UNSC weapon.2. Co-op with friends
If only one of you needs the achievement, then this has all the benefits of doing it with a second controller. The person who needs the achievement will want to hang back so as not to get killed.
If you all need the achievement, then I recommend putting the Iron Skull on. That way if one of you dies it will revert to the last checkpoint.
If you die you can pick up your weapon from where you died, but only if you can find it and it hasn't already been removed from the game or you didn't fall to your death. You can ask the host to reload the last checkpoint but if they are not quick enough you may get another checkpoint. You would then have to restart the mission from the beginning. I'll let you decide if it is worth that risk.
Alternatively for 2 player or 4 player co-op, have half the team hang back and keep both UNSC weapons. Try not to get killed, but if you do have the host reload the last checkpoint. The other half of the team can use whatever two weapons they want. Then just before final section with the ghosts, the ones at the back can hand over one of their UNSC weapons and everyone will get the achievement. Thanks to DubC8944
for this suggestion.3. Solo
If you die while playing solo you will revert to the last checkpoint, so as long as you keep one of your UNSC weapons, you will not lose it when you die. If you drop your UNSC weapon and cannot pick it up again, you can manually reload the last checkpoint as long as you still had the weapon at that point. The down side is that any progress you made since the last checkpoint will have to be repeated and once you have used up all the ammo in the UNSC weapon you will have to survive with just one weapon.