well I'm flabbergasted!
Have you asked DD what she wants? Do she and her siblings want to leave all their school friends that they have known, probably since nursery school to move to an area where they do not know anyone, to be pushed into a school that they are not familiar with? Has she said that anyone is making her life difficult because of her "genius"? Has she said she does not want to go to your local school with her friends?
For those boasting of their familial academic credentials - my father was an obstetrician. I and my 3 siblings were bright. My oldest sibling was considered a "genius" because of her amazing results, voracious reading and her ability to absorb encyclopaedic knowledge from the age of 7 onwards. She passed the 11+ but did not get into the school of her choice. She went to the local comp, where she was treated as the next Einstein and went on to get very poor A levels. Turned out that the "genius" was in fact Aspergers syndrome.
My brother did get into a very prestigious grammar school, to much fanfare and parental boasting. He had a thoroughly miserable time, being small and un sporty. He was the only pupil from his primary school there, so had difficulty breaking into new friendship groups. He was exceptionally bright, but got "eaten alive" because he was not one of the "lads" and the school was an all boys school.
My younger sister and I were the family "thickies". We got sent to the C of E Secondary Modern so as not to be an embarrassment to the other members of our family. It had no academic record whatsoever. We both agree that our time at school was the happiest time of our lives. We are the only 2 that went on to university. I am now a lawyer at a very prestigious law firm.
It does not matter how impressive the school is, how great their results, how fashionable the school is ( or the Kudos it attracts for the parents if the child gets in), children only thrive and achieve their full potential if they are happy. They may be bullied into getting good exam results by ambitious schools, only interested in their position in the league tables and equally ambitious parents, wanting to trump their associates at dinner parties, but at some point, the child becomes an adult and will have to succeed on their own merits and motivation.
My son is 11. I agree with the comments made by others regarding Milton Keynes schools. We came here 5 years ago when he was failing at his local, fashionable primary school. His new primary school has brought him right up from complete failure to at least level 5 and most importantly of all, he is very happy. He is going to one of the very well regarded comprehensives, with his friends and he can't wait to start (oh - did I mention, his dad and Grandfather went to Cambridge, his mother went to a top red brick, his grandmother was a solicitor, his granddad a prominent politician and of course his other grandfather was an obstetrician?)
Put the child first, parental ambition second and the rest will all fall into place.