A bigger school might mean more potential friends and more ways to get involved: sports, clubs, hobbies, etc. I think he would be much more likely to find some kinds that have something in common with him there.
If he is already unhappy and not caring at 10, I would make the change now. The sooner, the better in my opinion. Especially if that is what they both want to try.
Whatever you decide, I would still try and find something for him to get involved in, whether it's a sport, book club, or even picking up a hobby like model airplanes. It really doesn't matter, but it sounds like he needs something that interests him to care about.
Best of luck!