I am not sure I understand the state/private posts.
Well regarded specialist Universities like LSE and Imperial, as well as the maths/economics/science/engineering departments at other similarly regarded Universities, are hugely international. (The LSE's student demographic is roughly 25% UK, 25% EU, 50% international.) Little scope therefore, for the "what school did you go to" Top Trumps.
These Universities/departments will be rejecting qualified applicants. DS received three rejections even though he had a 4A* prediction. They want students who will thrive and who will contribute to University life, including its diversity. They will have data on how students from different backgrounds fare, and will, I assume, use contextualisation when deciding who to accept. DS' four closest British friends at LSE came from state schools, one grammar, three not. All four did just fine. Their international friends also came from a variety of backgrounds, including some on scholarships from their Government and another who had been to a posh UK boarding school. Yes there will be cliques, but equally plenty, from the whole range of backgrounds, who are happy to mix. (Oddly it was the French, rather than Chinese, who were perceived to stick together most, partly because Chinese students come from such a wide variety of backgrounds/places.)
Op's daughter would not have had an offer from LSE if she were not good enough. It is a University that never goes into clearing, it does not need to, not least because unlike oher Universities, they decided not to expand UG numbers. The LSE offers a world class education and she will do well, as long as she works hard, engages with her course, and is happy.
The latter is crucial. DS thrived there. He loved his course and met lots of like minded people. For him, this was the student experience he wanted. But he has little interest in rowing, May Balls, formals or clubbing. However he knew others who would have been happier elsewhere, and who felt that they had missed out. That is the decision she needs to make.