Having seen children being asked to leave one of these schools at 4, 6, 7 and 9 as "this school isn't right for them", I would actually be feeling quite relaxed about your decision.
We pulled out of the process before my eldest even sat the assessment as it was stressing me out so much. Then that decision stressed me out. Then we didn't get a place at any of our four state choices due to a change in the school system in our area. We got allocated an outstanding school but I just didn't like it. Overly religious, no outdoor space, gloomy buildings, no real fun about the place. This had me in tears.
Then we found a prep that blew us away. Not because of the results, although they do consistently send children to grammars, top independent and boarding schools, but because the kids seemed genuinely happy there. They overflowed with enthusiasm, looked after their peers, the staff talked to them like people not just children, manners are strictly enforced and they celebrate achievement in all areas not just the academics. Several years on and she is doing exceptionally well. On track to get into highly selective schools at 11 (Habs, NLCS, St Albans) and hey - no additional tuition required. She adores school, has no issues with stress and has a pretty fab extra curricular life. Some children leave at 7 to go to these schools but on the whole they stay until 11 and move then.
I guess what I am saying is that it sometimes taking a step away from what you think is "right" makes you realise it's perhaps not as perfect as you first thought. I do believe that if your child is happy and a confident learner then they will do well. These are the foundations that a child should get in the early years which they can build on. Private is not always the best, particularly in the early years.
I would also consider moving her at 11 instead. I am not in London (just outside in Herts) but have friends stuck in the whole 4+, 7+, 11+ cycle and it takes over their lives. Assessment and tuition is just part of their everyday family life. You CAN step away from it and the world will not end. You just need to be brave!!!!!
Whatever you chose - 7+ or 11+ your daughter will be fine. It's only three years you're talking about. She won't miss out on anything in the meantime.