You talk about pistol rounds in a rifle. Now to me a rifle is a rifle and is going to have to fit the boy. It is the weight and how long the barrel is ya know.
So if he is able to handle a rifle chambered for a pistol round then a Remington Model 7 in 243, 7mm08 would work.
I would have said a 260 also but the ammo is ungodly high priced.
My self I would not get thre youth model I would get the adult model and use a razor saw and cut thre stock down to fit than when he grows and has really long arms just add ther chunk you cut out of the stock for him.
Al