Putting Biological Father on the Birth Certificate Instead of the Husband

My question involves paternity law for the State of: Texas

can the biological father be put on the birth certificate at the hospital if the husband signs a denial of registrant’s paternity at the hospital?

Presumably you mean "Can the biological father and the mother execute an acknowledgment of paternity, so that his name goes on the birth certificate".

I don't see why not. What did the birth center say when you inquired about its policies?