Just curious, but, what military? Singapore, Aussie, ect.
Jesus, what crap. I worked with the Aussies a lot in Kandahar. Never heard anyone mention this. Doesnt matter, it is what it is.
Teaching combat shooting has some basic principles but each individual will have to find their own method. I understand your problem. A couple of years ago, I caught some minor shrapnel in my right knee and the right side of my face. Nothing serious, but, my right eye and cheek were swollen for several weeks. Since I was right eye dominant, this was a wake up call and I decided to teach myself to shoot with my left eye only. Now, I had an advantage, since I was using an M6 and not a bullpup, which is not my first choice in a CQB carbine.
Pistol was not problem, I just turned my torso slightly to the left and canted the pistol the same way. Carbine was different. I was medium accurate shooting from the left shoulder, but, way too slow for combat shooting. As Basement_Gainz correctly pointed out, cheek weld is vastly important for accurate shooting. The key word here is "accurate". For precision shooting past 50 yards, it is a must requirement (IMHO) but for CQB ranges at 50 yards or less, or clearing a room or being ambused in a vehicle convoy, you dont need to be that accurate, you just need to get a bullet somewhere in that body. I found that if I moved my carbine stock from my shoulder three or four inches to the left, and jammed it against my plate carrier, my left eye fell in line with the optic. Something to think about.
Having every recruit shooting right eyed/ right handed is just plain fucking stupid.
I am going to post a few links to articles and one to a forum dedicated to bullpup shooters. Hopefully, this will help and like Basement _Gainz said, check with your recruiter, this smells.