Most of the rifles listed are gonna boot you harder in the shoulder than a .243...even a lightweight one.
If you want to reduce recoil so your boy can use them your probably going to be looking at using lighter weight projectiles, a heavier rifle and/or putting a silencer on to reduce felt recoil.
Any of the cartridges based on the 308 case (243/260/7-08/308) are going to be readily available, reasonably tame on recoil with the right loads/rifle, suitable for a short action/barrelled bush gun.
Out of the cartridges you mentioned, the 260 would seem the sensible choice and would also be quite versatile too.
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