Is it possible to acheive Expert Marksmanship in BCT with cross eye dominance?

Definitely. I did it

SPC (Join to see) its possible. Many shooters close one eye when they shoot. Find a local rifle range and shoot daily. Make sure the weapon has been sighted properly

It is entirely possible. It is more difficult, but some things in life are. Adapt, improvise, overcome.

It WILL be helpful, though, having already identified a problem, that you communicate this to your Drill as soon as it is relevant. It is actually more common than you think, and your DS should be able to appropriately give guidance regardless of the cross dominance.

SPC (Join to see) Walking proof. Right handed, left eye dominant. You have to make up your mind quick. Do you force yourself to shoot left handed, or force your self to go with the weaker eye? Experienced NCOs told me that what ever you can achieve a steady hold, good body position, good trigger squeeze, and consistent sight picture is what you do. They can adjust your sights to make it work.

I tried shooting left handed....feels backwards. So I shoot right handed with my weaker eye. Exception is pistol. With practice I shoot well with pistol aiming with my left eye, both law enforcement tables and military tables.

The world is essentially right handed. The brass ejects into you, although the M16A2 and M4 eject forward and right. So you don't have to worry about hot brass down the front of your IBA. When my Dad went to basic training in 1968 they quarantined all the lefties. Buttoned the top button on the fatigues and put them all on one end of the firing line. A left handed drill sergeant worked with them to ensure they qualified. The original M16 ejected right but could flip hot brass down your shirt.

Trust in the training. Civilian shooting has little to do with military qualification tables between distances, timed courses and mandatory firing positions. You'll be shooting one type of ammunition. You'll be using adjustable iron sights. I don't think optics are usedmin IET. You may be issued an optic at your unit.

I am left yet, right hand dominant. Never had a problem. Just practice before hand.

Get in touch with a local gun club and find out if any NRA instructions are available in your community. There's a gun shop in my city that offers shooting and safety courses right at their site. Police, Air Force people, National Guardsmen, etc all go there for extra practice.
Since you are experienced, maybe just a couple tips will get you straight with it.

I am also cross-eye-dominant, right-handed but left-eyed, but I have no useful vision with my right eye. I shoot a pistol with my right hand, but I have to shoot a rifle left-handed. It is all a matter of training.

I am cross eye dominant and achieved expert marksmanship (in the USAF) without any special practice. Follow your DS instructions and remember that you are at your most stable with empty lungs.

Yes. As we transition to using optics as the dominant aiming device, this will become less of an issue. I am also cross-eyed dominant. I struggled with qualifying early in my military career. Once I understood that I was cross-eyed dominant and how to account for it I was able to do quite well with marksmanship. I'm currently on the Army reserve marksmanship team where I was able to earn the presents 100 tab and distinguished riflemans badge. I would highly recommend learning to shoot with your dominant eye. It does feel awkward at first, but most people quickly adapt. Learning to be ambidextrous helps substantially with the development of later marksmanship skills.

I did it but not until after basic training. My trick was to close my dominant eye just long enough to gain focus from my non dominant eye. Other's have said they fired from their non dominant hand but that's not the method I used. Good Luck.