This choice isn't just between grammar and independent, but also between single-sex and co-ed, a day school which empties in the afternoon and a boarding school, and as noted above, the schools have considerably different ethnic intakes.
I am concerned about the levels of state school funding at present, and how grammars will cope with even thinner budgets going forward. The schools have to either to continue to expand their pupil numbers whilst replacing more senior and expensive staff with younger less experienced staff, or chase external sponsorship/funding, or aim to get more pupil premium students in. You'll have read up on the recent changes with Nonsuch and WHSG forming a MAT. I feel that the changes which will result from that are still at an early stage. I would definitely be keeping an eye on recruitment - it is becoming more common now for even grammar schools to look for general humanity teachers rather than specialist teachers.
But fees are not insignificant, even with a scholarship. There are lots of opportunities at BC, but only you will know whether your dd is someone who would thrive there or not. She's clearly bright, so as long as she is happy and motivated she'll probably get similar academic results at either. You'll have had a chance to meet senior schools at both - I'd guess you would have a gut instinct at which your dd would be happiest.
That said, the schools are some distance apart, so you'd be factoring in either a house move or boarding I assume (or a heck of a long commute on Southern Rail!)
In terms of the wealth of the pupils - most of the time they're all in school uniform doing similar activities. Everyone is aware that any pupil's individual wealth is not as a result of their own merit or activity. Some families will be there as a result of "new money", some "old" and most trying to prioritise the fees alongside everything else.