A family bible is a book to record births, deaths, marriages. My granny has ours and it goes back about 300 years. No idea why they think it would have any impact on their plan.
Alternative forms of documentation, basically. Could probably get your child into elementary school and they would be on her record from that point onwards.
If not taken away by the social services, how would a child subjected to all of this work to gain normal status. Sounds like a setup that would result in no proof of American citizenship for the child.