Difficult.
I would expect choices at trad private secondary schools might limited, not least depending on whether they have spare places, and that your daughter might find subject choice restricted or she might have to repeat a year. However the schools you mention are nice and I have met parents from all who are pleased with them. You might however note that it is not unusual for bright girls from the smaller schools to seek a move at sixth form, especially if they want a wider subject choice or to do science.
An alternative might be to look at the sort of colleges which appeal to London's mobile international community. Britain's Universities and independent sixth forms attract huge numbers of students from abroad, and a significant proportion will be looking for a head start by studying for GCSE in London. A sector that at one time seemed to be the preserve of kids needing to resit, is now catering to a bright eyed, motivated and hard working group from all over the world, and increasingly offering GCSE classes.
One example is Ashbourne International College. The boy we knew there is switched from a non English speaking school environment elsewhere in Europe and heading for an English boarding school and had to work extremely hard to achieve a good batch of GCSEs in a year. Lots of praise for the college and also for its attempts to lay on some extra-curricular on top of a focussed learning environment.
Not what you might have been thinking of, but an alternative to a school where your daughter is not happy. You might find it easier for your daughter to settle where there is not already fixed friendship groups, and a couple of the schools you mention are quite small.
Portland Place has a similar history in that it started offering 11+ education partly to cater for ex-pats arriving with children in those difficult to place years, and will be more "school-like" and there are a surprising number of others.
If you were to decide after talking to the schools that getting GCSE out of the way at a college and then seek a place for sixth form is the most practical approach, the schools themselves might help you identify which of these colleges has a good reputation. They might also advise on whether the most practical solution is to simply grin and bear it for another year and a half, and then switch for sixth form, though you and your daughter would need to judge how damaging that would be.
Good luck. It is so miserable when your child is unhappy, and so important for her to see there is an end in sight.