A lightweight 243 is not any kind of low recoil. It uses twice the powder of low recoil rounds. If you have to have a traditional deer cartridge, find a reloader friend and a box of Speer 130gr for a 30/30. That's about it unless you get obscure with something like a .250 Savage, which by the way is pretty much a 6.5 Grendel.

6.5G kills better than 243. All the reduced recoil loads of common deer cartridges come back to 6.5G level performance. There is no free lunch. 6.5G does have cheaper target ammo available, but again while it is better than 223 it's not by a whole lot.

As for AR 22, that'd be good just don't think he'll be able to run it himself right off. About the only repeater a 7yo will run easily is a lever action.