King Edward School, in Edgbaston, by Birmingham Uni.
This year the head has managed to increase number of bursary places greatly. 25% of the intake this year, and next year, will have a bursary - some full, some partial, depends on your household income.
My youngest DS is a pupil there, on a full bursary, I cannot praise the school highly enough. Has lots of sports - all private schools do - but my not sporty son doesn't seem to have missed out at all: he does some sports (volleyball and athletics and swimming, just not in teams), martial arts, school choir, chess, book club, debating society, history society, maths society.
The KES website gives some details of the bursary scheme, and if you phone you will be given plenty more information; I found the school to be very helpful when I applied on behalf of ds a few years ago.
There is an entrance exam, held in Jan, I think; also think final date for submission has already passed for this years exam though. How old is your ds?