I hate to say this, as it's almost sacrilige on this website, but I think you would be better off buying a cheap little .22lr, specifically a bolt action one, and about a gazillion rounds of ammo.
If your goal is to become a good enough shot to qualify for sniper school, spend some time where it counts. Marksmanship.
You'll need to be a capable shot on more than just an M16, so forget about getting a "Kewl" rifle and just get something that you can shoot acuratly...ALOT.
Given a finite cash flow, you'll do better at bootcamp if you practice breathing,trigger squeeze, and conistant check weld with a .22 and a lot of bullets, than if you bought a cheap AR.
If you have unlimited funds, and can afford to shoot 5.56mm in great quantities, then sure, go that way, but it sounds from your post as that might not be the case.
The rifle doesn't make a good shooter, practice does. Physical and mental.\
B.R.A.S.S. - "Breath, Relax, Aim, Stop...Squeeze!" That's what will get you to qualify "expert", not some fancy, shiny, nifty AR-15, which, as stated above, won't have a whole lot in commen with the ratty, worn out rifles you'll get in fruitcamp.
If you aren't a good enough marksman to shoot that well at this point, it's going to take practice to get there, and the best way to do that is to learn some good marksmenship habbits and PRACTICE them.
That's why I say get a bolt action .22lr. The ammo is cheap, so you can shoot lots. They are accurate enough for training well, and you won't have to worry about learning bad habits with a semi-automatic rifle. I've seen that too many times. "Oh, I flubbed that shot, no biggie, I have 29 more...28 more, 27 more...Damn. Still not hitting right. Maybe I need to shoot faster..."
This is just my oppinion, but if you were my kid, I'd beat it into you. Shoot. Shoot lots. Shoot correctly, and hit the target! Let Uncle Sam buy you the M16. He gets them cheaper anyway.