Anec-data, but I've seen pro-lifers both for and against birth control, so perhaps this is some variance here.

In any event, I don't think being against birth control suggests the belief that a fetus deserves the rights of a child is necessarily implies the belief is in bad faith. Or at least, no more so than believing that murder should be a crime but supporting policies that indirectly lead to more murders (e.g. supporting policies that increase poverty or gun violence).

The more interesting gotcha for me is when pro-lifers concede an exception for rape. I don't think they believe a child born of rape is somehow less deserving of life than a broken condom. So that suggests the belief is less about simple life and death and more about a person's sexual choices.