Hello, I'd say that although what you propose may be perfectly possible it will not be easy and I wonder if your expectations are a little high?
Firstly, you've chosen one of the most popular school areas in London.
Secondly, entering at Year 2 is very common especially in London, however it is a non-standard entry point. I'm not sure if you realise, but most independent Schools in the U.K. have a Pre-Prep department which is years Reception to 2 then a Prep department which is usually Years 3+, so the normal intakes of pupils would be by assessment or exam into years Reception and 3. Therefore if you look for places in other Year groups you will either be relying on being at the top of a waiting list and a pupil having left, which admittedly happens fairly often in London compared to the rest of the country.
Due to the above, I would be surprised if you actually ended up much / any choice of schools for 2018. I think it will be more a case of where has spaces. For this reason, I'd advise having a State school option as a backup just in case. This will essentially involve picking a school and then moving house VERY close to it, however if you want to do this the details are complicated and I'd advise starting a post asking for State school admission advice for the area.
Can I ask, what qualifications do you want her to end up with long term? You mention she is at a European International School in the US, can I assume from this that if she were to stay there she would be taking the Baccalaureate? What are your long-term plans for country of residence? Are you hoping to stay in the U.K. permanently? Would you consider an international school here? Or a British school that teaches the Baccalaureate?
Also, you mention she is active and fidgety. Does this cause her difficulties/ could you see it doing so in the future?
Probably a stupid question, but are you really wedded to that location? Schools in the U.K. vary wildly. At many London schools she would only get 1.5 timetabled hours a week of fairly standard sports. At some schools further out of London children would do sport every day for an hour with 3 hours on Wednesday afternoons and occasional matches Saturdays, or instead perhaps 2-3 afternoons a week of sport plus Saturdays. Often 'Forest School' which is like Scouts. The range of sports available as after school clubs would be much more interesting too and could include things like shooting, horse riding, golf, archery, sailing, rock climbing. Could that suit her more?