@madeittofriday
Well this could have been me. I was awarded almost full scholarship for Y7 private school, with the alternative school both my parents taught at. They decided not to send me to the private school mostly because they were worried about paying my DB fees later.
I resented my DP, absolutely hated having them teaching at the same school, was picked on by other pupils and some staff who did not like my DF in particular. I stopped trying to do well and spent as little time at home or school as I could.
I applied again for Y9 entry and got a partial scholarship, smaller than before but everyone had realised it was for the best and off I went. Best thing that ever happened to me. My DB was perfectly happy at the local state school - had none of the same issues and we're now in our 40s with no issue at all about the different schools we ended up at.
Your DD sounds very bright and I would give her every chance to take the opportunity she has made for herself.
This.
Although my story is different, it may also help. My parents didn't let me sit the equivalent of the 11+ where I grew up (ideological reasons, although I doubt they could have afforded it either - left wing academic parents) and sent me to a local "very highly rated" comprehensive (although they picked - I was not consulted). I was absolutely miserable there, unbelieveably lonely and picked on for getting top grades - the school, for the first time, had a "girls" and "boys" science prize, because otherwise that would have meant a girl getting the top prize - which I think was their attempt to head off some of the bullying. We had ridiculous situations like me getting 100% in maths tests and the next top mark was 60%.
The only thing in my favour was that I was technically too young for the year - and age wise, fell within the age criteria (just) for the following year's 11+. So I pursuaded my parents to let me sit the scholarship exams (because it was based on age, not year), for an academic private school - we phoned around the schools to check that if I won the scholarship, they would let me go into Year 8, not Year 7, and while most said no, one said yes, I sat the scholarship, won it, and on the basis that I had "earned" my way into the school, my parents let me go. Undoubtedly the best thing that happened to me ever. I don't think I would ever have forgiven them if they had insisted on my staying for another 6 years of purgatory in the school in which they enrolled me.
BTW, after my experience, my parents let my DB sit the scholarship exams for the equivalent boys private schools. My DB won the scholarship - and decided to turn it down in favour of a new state grammar that they were just setting up (it was not selective in my time, three years later, it was). You never know what siblings will or won't want (or what opportunities might be available to them that were not available to the older ones). But there was no question that if he hadn't won the scholarship, they would have sent him to the private school anyway, just because I had. In their view - it was like I had financially contributed the amount of the scholarship (which in those days was full, and not means tested). In a way, it was not their money to throw away, it was effectively money I earned. Just as I would not take my adult child's earnings to subsidise my younger children, it would not have been right to take away what I earned just because my DB might or might not earn the same.
I think, given how much her heart is set on it, you really must let her go.