*Words of wisdom from another Grandmother!
Don't waste your money. A bright child will do well anywhere and when it comes to university admissions, state school kids often benefit from contextual offers as their achievements are rightly recognised.*
This is, unfortunately rubbish (I wish to the bottom of my heart it were true). Some bright kids also may have other personality traits that mean that they will do OK anywhere. Many do not. They may well not get anywhere near university if the mental health issues from bullying and/or from being bored and frustrated overwhelm them during secondary school. You may be lucky enough to have grandchildren that fall into "would be fine anywhere" category, but the generalisation of this is breathtaking. It also of course depends on how bright is bright and the profile of the others in the school. Bright can mean just a little bit more academically able that the pack at the school, or bright can mean getting 100% in every maths test - when in some cases the next top mark is 60%.
Even when you have a maths teacher that is thrilled by this (and this is by no means guaranteed, there are even maths teachers that can feel threatened, but I was lucky enough to have one of the thrilled kind), and keen to give you advanced work, keeps you busy and try to look after you (and my Year 7 maths teacher was fantastic, I kept in touch with her long after I won the scholarship and moved to the private school) - that makes you the teacher's pet and means you are even more isolated from your peer group. It is lose-lose.
Clearly none of your grandchildren have spent every break in the library (because one was allowed to escape the bullying in the library) pulling out every book in the library starting from A, and reading at least the first few pages to see if they were interesting. Sure I was exposed to an extraordinary range of books that way, you can get through most of a school library like that in a year. But what I would have done to have swapped it for a Year 7 in the private school I eventually went to. I used to come home from school pretty much every day and go into the neighbour's garden (her daughter had grown up, and we had use of it) and cry - sometimes for hours at a time. Completely stopped once I went to the private school.
We don't know exactly what type of child the OP's DD1 is - the first or second kind. But the best guess is that she is the second. Why? She has been driving this! Kids who drive with this kind of intensity often know something about themselves that others don't. When I came out of the scholarship exam, my DM said that as she was waiting for me, she was watching all the other little girls coming out, and she wondered if any of them wanted that scholarship as much as I did. The OP's DD is pushing with all her might towards the independent school. That tells you an awful lot about what she thinks awaits her at the alternative. At 11, a child, and particularly a very bright child, is likely to have at least some insight into themselves. She could of course be wrong. But I suspect if she is wrong, and wants to then switch to the OP's school, that will not be difficult. Tell then the following year thanks but no thanks. Switching the other way will be impossible.