Hello there
First - congratulations! Bousfield is one of those near impossible state schools to get into.
I wanted to add that a lot depends on where you are planning to live in the next several years and what path you'd like your DD to take.
If you are going to be relocating again in the next several years, than 11+ exams that private preps are working towards are not relevant for you and you don't need you DD to be going at a faster speed in the early years, which is what top preps would do.
If you think you are staying in the UK and 11+ is in your DD future, then your choice between Private/State might be different. Which, btw, still might not be in favor of Redcliffe.
As I understand, Bousfield has a good academic reputation among state schools in this area. Redcliff, on the other hand, is a small, and happy, but not a hugely academic school. They don't even publish leaver's destinations by year, so it's hard to compare them with other schools.
If your DD is focused and academically minded (as much as a 4 yo can be...:) ), then if I were you, i'd wait for an occasional place at the more academic schools in the area - Glendower, Falkner, Ken Prep (a bit further, though). Every year there are "no places left" at around this time for entry to Reception, and then people move around and places open up. And you can always move from State to Private.
I guess, my advice would be to take Bousfield now and figure out your priorities/goals for the future and adjust accordingly. It is surprising to me as I am a parent of a DD in a private school in the area...