Many religious objections to birth control consist of "It's actually abortion, just a bit more subtle" - preventing implantation of a fertilized embryo is the same as a surgical abortion, if you don't distinguish between a day's gestation and two months'. Most of the rest seem like generalized objections to sex - human biology being what it is, the punishment for that will inevitably fall largely on the shoulders of women. It doesn't seem like it's just a female-specific objection, though - I doubt your average religious objector would get too worked up at the thought of alimony or a shotgun marriage, and most seem to actively encourage adoption.

As for your proposed strategy, it seems like it's basically trying to do the same thing, given how liberalized sex and liberalized religion are so tightly bound in practice.