OP, I think you're overthinking this a bit.
If the school knows your religious orientation and is happy to accept your child, how are you being hypocritical? As long as you are happy to accept and support the general ethos of the school, I don't see any issue.
Like posters upthread, I wouldn't want to suggest that a religious upbringing is the only way to instal values in a child (although it is true that religion is a well-trodden path, and that values in the UK are by reason of history developed by religion). However, it is ridiculous to suggest that exposing your child to ideas about God is in some harmful, as if akin to exposing a child to pornographic material. The Church of England takes a very nuanced view on a good many moral issues and, regardless of whether or not your DD decides to believe in God there is every reason to believe that the school will teach her to deal with such issues in a sophisticated way. The CofE is not so much liberal as intellectual and respectful of disagreement in this respect - far more than other churches.
Also, learning something about Christianity in a meaningful way helps understand some of the traditions of the UK and also traditions of other (generally more religious) countries outside the UK. I shake my head at some of the absolute guff people come out with, for example, references to the theology of the Old Testament - which demonstrate no idea of the principles the churches use to handle it.
Finally, you mention being troubled by modern Christian teaching on women and sexuality. I think the better question is to ask whether you are troubled by CofE teaching on those subjects.
Yours sincerely,
Toad (child of one Anglican and one atheist, taught with his siblings at a CofE school, remains Anglican, although not all his siblings do, and was confirmed at the church of an openly gay vicar in the 1990s).