The .243 can have enough recoil to be too much for some smaller shooters, however, many can handle it just fine.
You know your son better than we do and what his pain tolerance and other nuances are that could affect his handling of a firearm.
My younger brother could not handle a .243 until he was about 13, but he is small for his age.
I would go with either the .223 or the .243. The benefits of the .243 are some more energy going downrange. The benefits of the .223 are that it is fairly inexpensive to practice with, and has less recoil. Of course, if you reload then price difference will be negligible.
The .22-250 just seems like it has the cost of .243 ammunition without the extra "oomph" and the recoil of the .223. An overpriced .223 if you will (for your purposes).