Geno...
I'm the same boat with my 7 year old also. He can shoot the irons on the .22 lr with in a soft ball at 50 yards no problem. I tried to start him on a youth model .243 in prep for upcoming deer season... Two steps backward. Here are a couple things I learned:
Chiller is right. Higher mag is not your friend. Their eyes are much better than ours. They can center a fine cross-hair on low mag no problem. Higher mag will make their positioning behind the scope more critical and magnify movement, and if your kid has a short attention span like mine, he will get aggravated real quick. Make the stock (cheek riser) fit him so he developes positon memory and doesn't have to screw around with modifying shoulder and head placement for proper eye relief. The higher mag will also bring out mirage quicker. If your son understands how to deal with it, no problem.
I also tried a 3-9x and had to go with something that allowed me dial out parallax. My son didn't understand "consistent sight picture" like i do. When we started practicing at 100+ yards, his parallax issues magnified his misses and he got down on himself. Then again, if your son understands sight alignment and sight picture, then a non adjustable parallax scope will also make him work harder to achieve consistency.
I don't know how a 22-250 recoils, but the jump from .22 lr to .243 was too much for my son. The increase in noise and recoil made him develope a nasty flinch quickly. I have since gone to a .223, and even reduced loads on my .308 with a suppressor to help with getting used to noise and recoil. I have also put a scope on a .22lr that allows me to adjust parallax in hopes that we can learn positioning behind a scope.
Right now we are keeping things within 100 yards and hitting big fat balloons. He has fun. I have started making the balloons smaller, and he is still nailing them. Next I will move them farther out and make them even smaller.
He sounds like he is progressing faster than my boy. You know your kid better than we do. His actions will dictate how fast you train him. But I would definitely get a scope with adjustable objective or side parallax if he is going to be shooting longer distances. Wouldn't you want one??? As far as FFP or SFP, I can't really comment as I have only shot SFP.
Nothing wrong with a higher mag range on the scope, but I would make sure the low end is no more than 3x to 4x (ie 3-12, 3-15, 4-16, 4-20...etc). Cranking up to 22x or higher can be used as a spotter so to say, but definitely teach him to shoot on the lowest power possible.
Baby steps.... and lots of patience. Be sure to post pics.
Enjoy every minute of it.
Bill