and plenty of smaller private schools will have connections to livery yards etc.
TBH its more a case of pick schools you like and then ask about taking a pony - no point in going to a school just because it takes ponies and hating it.
It's not quite south east, more west, but I agree about Stonar. Our daughter goes there, it is an excellent school and the equestrian side is fantastic.
Stonar thirded or fourthed or whatever. Super school. Not sure if there are others with actual yards, but if you google ISODE, you'll at least find the schools that entered teams and that's a start.