We have looked at all 3 schools as well. I’m not sure I would agree that Charterhouse Square is more academic than the other 2. I think all 3 have good teaching and bright children who do well at 11+ (or 13+ for St Paul's). But I think naus is correct that each school has its own ethos, and I would choose the one that fits best with your daughter and what kind of education you want for her.
Our impression (and this may be wrong) is that Charterhouse is very focused on exam prep, even from quite early on, which might leave less time for other types of learning. Perhaps consequently, music and arts are not a very big part of the curriculum, while they are very important at both St Paul's and Lyceum. This may or may not be something you’re looking for as a parent. Our son is not very musical generally, but we still think it’s important that he learn an instrument and read music, so Charterhouse probably isn’t a good choice for us.
The class size at Charterhouse is also very large for a prep school - 26 in a class - whereas St Paul's and Lyceum have maybe 18, fewer probably at Lyceum. And because of the class size, the Charterhouse building also felt very claustrophobic to us. The buildings at St Paul’s and Lyceum (their new building at least) are bigger, with science labs, music practice rooms and their own canteens. They also have their own outdoor play areas, though Charterhouse does have use of Charterhouse Square.
According to the head a few months ago, there were several places available in our son's year at Charterhouse, so I don't know how hard it really is to get a place (that is, if you don't get a place in the lottery for age 3) as people do move out, and I’ve heard some move from Charterhouse to St Paul’s for reception. But it may just depend on the year. We also have found that most who want to go to St Paul’s do eventually get a place at 4+ or 7+, but occasional places do not come up very often. Lyceum is growing as a school, and as far as I know it has places available in every year still (not that that’s relevant to you since your daughter is already there).
One other thing, you might want to ask MNHQ to move this to the primary education forum as it's possible you'll get more opinions there!