As a midwife myself, no, I don't think you're going to be able to blackmail stray children with the truth of their parentage. Considering the way of things, there's a decent enough chance they don't know their parentage to begin with. Unless you manage to uncover one is secretly the child of nobility, it's not gonna make much difference to the children.

Talking to the midwives can rule out that any of ours are their parents, though. That'll do for respectability and maybe get us off the hook for responsibility.