Our DDs are at LEH and like you I considered all three schools a few years back. IMO they are three quite different schools. LEH is single sex 7 - 18, Latymer is co-ed 7-18 and Bute is single sex 7-11, so this is not a comparison of like with like. Also all three are selective, i.e. to some extent, they choose you. not the other way around and all three are looking for different things. I have never met anyone whose dd got offers from all three schools, which maybe highlights that they are different to one another.
But I struck Bute off my list quite quickly. It was obvious to me on visiting that our family would not fit in there - lots of Mayfair, Chelsea types with girls being dropped off by drivers and the writing on the walls demonstrated that the pupils enjoyed lots of very exquisite foreign holidays in far flung places. Also the fact that my DDs would be thrown into the 11+ scrum quite soon after moving schools was not a draw.
Latymer - to me, it felt like the girls were 'allowed' into a boys school. It didn't feel balanced. I was put off by it's lack of sports facilities on site ( I know you can bus to White City). Also the whole entrance process seemed very formal and er, exam like and pressurised.
At LEH, it was geared to girls, of which I have a few, amazing facilities, including sports and the entrance process was very gentle by comparison. The girls turn up, spend a morning in a classroom, can put up their hands and ask questions etc if stuck, do some Maths and English, have a playtime outside, get fed a biscuit then go home. Every child gets an interview - i.e. group reading and a v quick chat about hobbies and that's it. Then they stay until 18. I am extremely happy with the school - academic, sporty, arty, parents in old cars who holiday in Dorset ( a few more wealthy, of course... but not so many) and very experienced staff who have seen it all. I'm a big fan.
Good luck with your decision.