Regarding Christian school, you need to be cautioned that there will be other parents doing the same thing (if the school allows students who are not member's of the church family.) That means that not only will the other children be learning social skills etc as any child that age, but they might come with some language from home that you wouldn't otherwise expect, and other such things.

I have been thinking about you and this... and wonder if you realize how much of your ethics and morals, that you don't wish to push on your own child, are what have made you who you are...and that you learned much of this from your Christian upbringing? I think if you don't give her a close to/similar upbringing that you won't be on the same page to begin with. . . you might consider recognizing your roots, your faith that you do own, and allow your child the same opportunity to get to decide her own spirituality. If you give her nothing, she won't find it.