Best you can do is either play their game, which does not look to be working out, or start a Record and begin the exposure process. It may turn out that they have the authority - but I doubt it! If you confirm their authority you will have done a lot more for society than anybody else has.

This delegation by the hospital and State to religious organizations comes off very irregular and peculiar at first blush.
Please excuse the absence. I had my leg ripped off in 09 and have been attending to my medical treatment plan. Definite goal of establishing a record. I've been doing it at the bank for over a year now and this case is a perfect vehicle to get my intentions on record.

I think it goes without saying that courts are grossly stacked in favor of the mother.
Trafficking in children is a cash cow for state (government) as well.

Your odds are <10% of receiving custody of the child as a father in (their) court using standard procedures.

In the future if you decide to marry and have children, I recommend going overseas. You still have to be careful, but I think you have a better shot of actually having a greater ownership interest in children rather than the state (government) owning your kid under the (old Common Law) doctrine of parens patriae
I've made certain Life choices that are now requiring responsibility. Perpetual traveling seems the way to go when my obligations are fewer.

find a woman who is into unassisted home birth/no vax/breast-is-best/homeschooling, etc; then gov. noseys don't go sticking their nosey noses in stuff.
It's hard to even get them to look up from face book now:/ Life choices. Be care full what you wish for because you just might get it?