I think its very strange that your relationship would be over if you didn't do this? You have every right NOT to give up the birth certificate, its not their child, surely you can make financial decisions for your own child?!?!?

If they are so keen on opening an account for her - tell them to open a separate one and "reserve" it just for her. Does it NEED to be in her name? Not at all, tell them to put it in their own bloody names, and just deposit money for her.

I'm sure they can be trusted to not touch the money themselves