Down wind,
A good .25 bore would be the old .257 Roberts this gun dose not get enough credit with todays ammo listing go with my favorite deer cartrige a 25-06 rem.I have killed mule deer and white-tail not a one has ever got away.

Ballistic,
The diffrence in recoil in the .243 and 25-06 is not noticeble,but I' m not 8.I have both the .243,6mm and 25-06,I would compare them all close to th .410 bore 3" mag.
If this is gun for him to grow in too,I would go 25-06. Buy a Ruger M77 MkII full size,then buy a Ram-lite stock and sawzall it down to his size and some 90 gr Barnes-X and shoot away.