If he signed the papers acknowledging paternity the only way to get him off is to get the bio dad to take a paternity test at least in my state through the courts. You need to talk to a lawyer bc you both falsified a legal document. Id tread very carefully. Changing the last name would probably require the same thing going to court
I would not recommend putting bio dad on there at all. File with the Court to have her name changed. Request DNA from bad bf if the court requires. Then tell them you don’t know who the bio dad is. They can’t make you “remember if you don’t.” Bio dad has no business being on anything either. If he wants to in the future then he can go ask the court properly.
Then it will be only you and baby on there. Take care of and focus on you and the baby and good.job getting out of a bad situation.