I'd say get a rifle he can grow into and use throughout his life.
The .260 Rem has very mild recoil but can be tuned into a 1000yd deer or antelope rifle very easily.
I went through something similar with a friend many years ago. He'd put both he and himself in for the draw for Elk Permits in NM.
They both drew up.
For him we bought a light sporter 700bdl rem in 7mm mag.
I had him cut the stock down to fit the boy, and save the cut off block. I also had him undercut by about 10 degrees shorter on the bottom so as to help cut down with muzzle flip directing more of the force straight backward.
I then worked up some very light loads using if I remember right 110gr sierra's.
The kid had zero problems handling the rifle and shot very well with it. I got a call after the first night of the hunt from the happiest kid in NM. He'd just taken his first game animal, and it was a very nice cow elk killed at a measured 327 yds.
As the kid grew up dad simply had the cut off piece put back on the stock, and the kid started shooting factory "normal loads".
Remington offers some reduced recoil loads I believe for the .260
and coupled with a nice muzzle brake
your kid should be able to handle them fine.
I would say look at one of the varmint packages as the heavier barrel will help a bit with felt recoil; although you may end up having to carry it for him.