I have never heard of such a thing... Biological father resides with girlfriend (fiance just recently) been together for a year and he wants to waive his parental rights to her? Is this even possible? But yet they want to go to court for full custody? Biological mother does not want this does she have more grounds that if he pursues this action can she not have his rights waived to her??

No, rights are not given like a title to a car. There is nothing he can do in court, mother needs to be laughing at how stupid he is. Is he trying to do this threat to get out of some future court action, or as a threat since mother is asking for something

He can't waive his rights period! He can not fight the mother on custody, but that's about it. A child is not property that it can be passed around.