My DDs were both at Berkhampstead from 3-11 (the youngest is 20 now, mind, so some time ago and things might have changed). It was a lovely school, well-known for its happy children - seemed to get the balance just right between academic pace and being a place where children flourished emotionally.
One of mine went on to a local independent school (DC) on a very generous music scholarship (the music at Berky was excellent) and the other to the local grammar. If your child is academic Berky has a very good scholarship haul to private secondaries every year (it also gives scholarships at 7 for both internal and external candidates) but quite a few leavers also take up state grammar places (there are several grammars in Chelt/Glos.)
My feeling is that Berky does (or 'did') well with both academic and not-so-academic children and certainly in my day the one word everyone used to describe it as a school was 'happy' - which is probably high on your list!
The ratio of boys is much better than it used to be - nearly 50/50 last time I looked - and they do their best with sport, although sporting facilities are not their strongest point (the site is somewhat cramped as you will see if you visit.) Music as I say used to be excellent.
However, if I had my time over again, I probably would go with the state option, personally; private schools just cost so much and state primaries in mostly-priveleged Cheltenham are often very good indeed.
For a good state primary, Holy Apostles not far from Berky (at Battledown) has fantastic Ofsteds and wonderful grounds for sport, with bright and modern buildings. I know it personally and would consider it well worth a look, if you don't mind that it is C of E with a 'strong Christian ethos.'