I'm right handed for most things, but not for shooting anything (tried right for a few years, but switched). Left eye dominant as my right has issues. He actually might find it easier because despite the handedness it doesn't really not work, with the exception of learning to feel the trigger. I shoot a RH shotgun as those have been hard to come by for lefty (its a little weird on the load, but actually works ok as I'm more dexterous with my RH so loading is easier, just rotate outwards), but shoot LH bolt actions. My wife really struggled with the whole LH/RH thing til we eventually worked out she was RH - left eye dominant. She's got similar challenges to your son. Switched her over and now she makes me look bad in 22 LR comps. Its probably worth a try - if it avoids him getting smacked with recoil from the cross over it might encourage him to switch over. If it doesn't work, you might want to look at a way to lengthen the pull on a RH rifle to avoid him crouching over it quite so much.
Definitely agree with the suggestions on caliber - 6.5 or 270 are a couple of options - a wildcard might be something like a real flexible option like a 6.5x55SE - which you can run a 140 gr in for a real soft shooting experience that'll work for any deer, or a 156gr that they use for Moose over in the Nordic countries that will work for elk at reasonable ranges. Ammo is pretty good on availability, and its my favorite rifle by far.