Practical thoughts; Have you looked at other sources of funding e.g. the Directory of Grant Making Trusts (all sorts of eccentric and very specific sources of funds in there, may hit your case, may well not, worth a look), also support from 'old boys and girls', any other charitable foundations?
Have you talked to the school about what score your dd actually achieved? If they go up to 100% bursary for the brightest, why are you not being offered 100%? Is it that she did very well indeed but not truly exceptionally? Or is it that they judge you could pay more than you think you can? Is it worth discussing that, if only so that you understand?
Have you looked at options from 11 and can you work towards a full scholarship, or bursary combination for that stage, if you think she would really benefit?
Can you think about how you can use any money you would have put towards the fees to provide extra-curricular activities, tuition, interesting days out etc? She could achieve a lot by being stretched in these ways, adding breadth as well as some depth, without seeing it as 'school', so possibly having more fun and feeling less pressured or competitive (and if you're going to say she thrives on being driven, well, you can do that too, through music lessons, maths tuition etc).
Essentially work with what you have a be positive, bitterness will eat you and doesn't sound all that well-founded anyway. Presumably the other family's bursary would not have been transferred to your dd, at best it might have been divided between all the bursary recipients, or, it would have gone to the next child down the list. The school has criteria, that family met them. The school is not omniscient, no-one is, nothing is perfectly fair.