CPS can take a child whether they have a birth certificate or not. However, I am now a foster mother, and I am learning that really, CPS is typically not malicious. Social workers really do want the best for the children. They work really hard to reunify them with their biological parents as well, even people who have had 7 children taken away... they still make sure they have supervised visitation, etc. So really, my viewpoint on CPS has changed a lot now that I am working with them. I am surprised sometimes at how hard they DO work to reunify, when personally, I think 'that drug using parent is a piece of crap, WHY would we try to get their kids back with them?' Anyhow, I know this won't make much difference with your fears. I had the same fears for years, and had a social worker friend tell me 'they really aren't trying to tear apart families', but it didn't matter. I do know that there are some bad social workers, just like there are some bad cops, some bad doctors, etc.
FYI, what we did to avoid issues with this is... after we started doing our own prenatal care, we wrote it all down just like the midwife's office had. We took him to the doctor less than a week after his birth for a checkup. We applied for the birth certificate with all of our prenatal care information and doctor's record of seeing him. If we had transferred, we would have been honest, but showed them all of our prenatal care that we had done, etc. Hopefully, even if CPS had investigated, they wouldn't have actually opened a case.