I knew I would see this. Please don't let him shoot anything smaller than a 243 even if it IS legal.
I second the opinions for wait til he can shoulder a rifle. Even if you have to cut the stock. I don't see anything good coming from letting him shoot from his armpit.
For decades, the poaching gun of choice was a .22 magnum because it was legal for small game and coon hunting. Countless deer were illegally killed with it. I would think that the .223 or 22-250 would more than match up with that weapon.
I would rather see a young person shoot a caliber that is light on recoil that he/she can shoot well than a heavier caliber that is more socially acceptable.