A few thoughts on your post:-
The GCSE results from KES seem to be substantially better than BV, with 46% A* compared with 14* at BV. 77% of KES A levels were grade A, compared with 42% at BV, and KES got 26 boys into Oxford & Cambridge, compared with 6 from BV.
Both are excellent schools, but I don't think you can read that much into GCSE A*s. Bright, well motivated boys will get good GCSEs at either school. The higher proportion of A*s at KES is probably more a reflection of the fact that KES boys tend to be more driven by their parents to achieve and overcome the natural lack of maturity in years 7-9.
A Levels is more interesting, with KES switching to IB I would suggest that this may not be that straightforward for some parents. My personal view is that IB can be a little boring and not necessarily the best way forward for those students wishing to specialise in science and engineering. If I had a boy at KES doing GCSE who want to follow a science or medical career I would give serious thought to moving him to a 6th form college after GCSEs.
DS is keen on music & sport, is an avid reader, and at the moment enjoys maths & science more than other subjects. He's at his local state primary at present.
KES is clearly outstanding for music, but part of the reason is the amount of time that boys are allowed to spend on music. BV doesn't enjoy that luxury having to follow the national curriculum. Having said that BV has now got Alison Timms at the head of music, she used to be head of Music at KEHS some years so I think pupils and parents will notice the difference.
Difficult choice, suggest you ask for a visit to BV with your son.
Hope this helps