Good day. This my first solution I am publishing so let me know if you have suggestions.
I played the rest of the game on Normal difficulty because I liked a little bit of a challenge but I was able to get this on Casual. I do expect that helped me a little bit. I also recommend setting the control-style to Pitch / Roll as the beginning of this mission requires A LOT of very slight adjustments to your rudder. The slightest foul up will fail the mission right at the start and it is a pain.
If you read anything in this, I mainly recommend reading the first objective walkthrough and the heads up/glitch alert at the beginning of the fifth objective walkthrough.OBJECTIVE 1: STAY IN FORMATION
This was the must frustrating part of the mission and it really could have done well without it. The rest of the mission is not bad. You are flying a B-25 Bomber in the very middle of a very crowded formation. The game didn't seem to be very consistent about how level you were starting out. Sometimes, I had to level and course correct almost immediately. So you need to stay in the very middle of the formation or center screen for this entire section.
You will notice that as your heading adjusts, there will be dark red arrows that fade in around your plane when you start to get out of formation. They are telling you which direction you need to head up, down, left, or right to get back to the center. You should be OK to keep your air speed between 90% and 100% during this section but dumping speed too much to stay in formation did not help all that much.
Use
and
ever so slightly to get on course. If you have only one red arrow showing up on screen you are doing OK. If you have two, you will want to start turning by holding
and
a little longer and turning the plane slightly left or right as needed with the
or
.
If you have three red arrows, you are in real trouble. Likely, you will be too low or too high in altitude compared to the rest of the formation. Move your left stick up or down as needed and speed up a little bit to get back to the center, and QUICK! If you get off course, there is a blue bar timer that very quickly counts down the amount of time you have to get back in formation. If you have the three arrows too long, this bar counts down 5 seconds that you have to get back. Also be careful, not to collide with the other planes!
The formation is going to bank to the right slightly at least twice before continuing straight onwards to the next objective. These turns are going to be the most difficult for you to course correct on. They don't seem like they would be very hard for you to manage but you'd be mistaken.
I feel like entering in a spoiler alert even though there is not really a story. If you would like to be surprised as to what objectives come next, please try them first.
*** Spoiler - click to reveal ***OBJECTIVE TWO: REAR GUNNER DEFENSE
There will be a lot of Japanese fighters that will come at you. I did not spin my turret to the left or right too much. The spin speed for me is not very fast and downright sluggish and you can't really hit the targets that are hitting you hard immediately behind the plane.
Unless you find real targets of opportunity, just keep your guns between the two tail rudders and unload your guns on the planes that are leveled straight at you or those that are behind the plane and are slowly straightening themselves out to shoot at you. Of course, aim for the "aim suggestion" reticles to take out your targets. As you may know, once your reticle is flashing bright red, your guns have overheated so only fire as much as you can and let off the trigger briefly so you can keep firing.
You only need to shoot down enough fighters without dying until your guns run out of ammunition as per the story.
OBJECTIVE THREE: FIGHTER ESCORT
Your P-40s finally arrive. There is a B-25 in the group that is damaged that you have to prevent from being shot down. It has a green bar that shows how much health it has. Shoot down all of the fighters attacking it.
I sometimes circled around the formation shooting fighters that were already smoking. Mostly though, I tried to stay behind the bombers shooting into the fighters that cluster in behind them. Don't worry about damaging the bombers if you shoot into them.
Once there are only three fighters left and the damaged bomber is still alive, they will bug out and the game will have you follow them and take them out.
You probably will not be able to shoot down any of them down, You might be able to pepper the one in the rear but they are just too far away. You will find out through the story that their withdrawal is a means that they are springing a trap.
OBJECTIVE FOUR: DOGFIGHT
After the cut scene, you will be at high altitude diving down on a flight of five Zeroes. They may be another opportunity for you to earn the Chennault's Pursuit Theory achievement.
I found that my AI companions took out 3 of the fighters within a minute; so likely you will just need to take out one plane. Circle around and slow down as needed to shoot ahead of the fourth fighter to shoot him down.
OBJECTIVE FIVE: TAKE OUT FIGHTER ACE
HEADS UP -- at the end of the dogfight mentioned before, make sure that you have plenty of altitude so that you do not crash. I found a glitch I think at the beginning of the cutscene revealing the Japanese fighter ace where apparently I must have crashed during the movie. Really strange. In fact, just skip the cutscene by hitting 
to make sure the mission doesn't fail by a crash completely out of your control.
This fighter ace in my opinion is the easiest to take out compared to the ones rest of the game. Unload your machine guns on him in head to head duels when he is flying straight at you. The aim ahead of him as he flies whenever you can as he circles.
The achievement should pop when the final cutscene starts.
After final cutscene ends, stick around after reading the epilogue, historical facts regarding the Flying Tigers and the achievment to complete the game will pop when the credits start playing.
I hope this helped!