A short note asking that, if possible, the two girls are kept separate is enough. Schools are used to it. You might also suggest a girl that you would like your daughter to be with so the note stays positive.
You have really been given the full Mumsnet treatment which I am not sure is strictly necessary. You could be right, and the girl is a nightmare. However beyond asking that your daughter is kept separate and being alert to future problems, there is not much you can do.
If there is a genuine problem, others will be doing the same. If the school gets several such letters they will know there is an issue they will need to manage. If you make too much of a fuss the school will perceive it as a problem between your daughter and this girl. You don't want this.
If they are kept separate there should not be a problem. Your daughter is forewarned and will know to steer a wide berth. If there are problems it is more likely to be with girls who are not aware of past behaviours. Giving the school a gentle indication that this has happened before will enable them to respond quickly, should, and this is a conditional, should things happen again.
Girls change a lot between the ages of 10 and 13. They settle down, find their own groups and start leaving others alone. They can also over-dramatise day to day conflicts. I wont be the only mother to have a DD who suddenly proclaims that the girl she has done nothing but moan about, is now her new BFF. (Luckily she has an older brother who rolls his eyes, meaning there is no need for us to say anything.) All you want, and all you can ask for, is distance.
If things are then a problem you can escalate. But keep your powder dry till then. And rely on the fact that if you have a problem others probably do as well, so no need to rush to the fore.
On bursaries I would say in your defence that at primary level kids boast to each other, so we tended to know who had bursaries at DCs prep. At secondary it is completely different. DD knows that one of her closest friends is on a 100% bursary, and DS might guess, but would be too disinterested to ask, that a couple of his close friends receive financial support. Beyond that we have no idea.
There were also a couple of rather tactless/insecure mothers who felt the need to tell everyone what reduction in fees their child would be getting at senior school. Given others are making sacrifices to afford fees, or would dearly love to send their children to an indie but cannot afford to, this can really grate, and can inevitably cause people to wonder why one child is more deserving than another. Truth is you are not privy to all the facts, and even if the school has made a wrong decision, there is not much you can do about it.