I've not shot the .243, but have shot a lot of 6mm Remington; the 243's ballistic twin. The recoil is noticeable and I'd not want to hand one to a sub-teen shooter unless I'd put together some lower powered handloads.
The 223 and the 22-250, otoh, are quite tame and both are very accurate. Cloverleaf accuracy is the norm with my old 788 Rem. in 22-250. Bought it new 45 years ago for less than $100. My 6mms are no slouch accuracy-wise, but neither come close to that 22-250.
My vote goes to the 22-250; especially if you can find one with a fast enough twist to stabilize the heavier 22 cal. bullets. Very satisfying to shoot at both targets and varmits without enough recoil to cause flinching or develop other bad habits. Let the kid develop as a shooter before trying to make him into a big game hunter.
If you manage to find a 788 (in any caliber), you can cut down the wooden stock to fit the kid and later buy a full-size plastic stock to fit him when he's grown. Nearly all of them offer gilt-edged accuracy.