OP, your DD is 11 years old. She has not one, but two, older siblings that have both got into this independent school and now it’s her turn to do the same. Of course this is massive pressure and, whatever she says, of course this is the fear of failure talking!
Also, you must know from the experiences with your older DC that primary friendships are generally irrelevant in that they change to secondary. This will be the case even if she goes to the state, but of course, your DD can’t possibly know that. She doesn’t have the benefit of hindsight yet.
Are you talking about Latymer Upper School, by the way (I only ask this because there are not that many co-Ed London Day Schools with those results)? I have DC there and, as I’m sure you know, at least 20% are on bursaries. There is a policy that no child ever misses out in a trip due to financial circumstances - this simply does not happen. My DC have friends from all walks of life - in fact, they have the chance to mix with a wider demographic than you wouid find in any state school. London independents are very different to being closeted away in the countryside somewhere, as I’m sure you know. So what does she mean when she says she won’t fit in? She comes from a family who are already part of the school! I can’t think of any school that has a “personality type.” If she’s worried about academic pressure, well, again, as you must know, these schools are self-selecting. It’s rare for a child who would not cope to be offered a place at 11 plus. The odds of a place are one in ten for some of these schools, so they have no need to take children who might not cope.
Yes she may well get all “9s” at a state school, but she will be one if a few if she does, rather than part of the norm. Don’t underestimate the impact of peer comparison throughout school. My DS thought he was very average all the way through, as did many of his friends tbh, yet they all came out with all 9s. Nobody rests on their laurels, in other words. Nobody can be complacent for a second. In a state school, they would have been constantly told they were way above expectations; top of the top sets; etc etc and wouldn’t have worked so hard. That’s what happened to me.
If your DD is doing 11 plus for this particular independent, is she not doing any other entrance exams for other schools? Even if she doesn’t get into this one school, it would probably do her well to feel that she got in somewhere, even if she eventually does end up going to the state school. Maybe sit the Girls Consortium exam as there’s not much you can do for this one anyway and there’s about 12 schools all over London you can apply for through it, some easier to get into than others?
I would say to her that she can’t possibly make choices until you all know what those choices are. I wish you the best of luck, having gone through this process several times. At least she has a good state option. This is a positive!