“Even if no beis din has so required?”
I don’t think he has any leg to stand on to say otherwise. Technically he’s within his (legal and moral) right to dispute in Beis Din his having to give it. But since he remarried and abandoned his first marriage I think it’s obvious Beis Din will require he give it. (Although, depending on the circumstances, it’s possible they’ll delay its giving until all other outstanding unresolved marital disputes are first settled.) So he legally is within his rights to first have a trial to determine his obligation to give it. But based on the circumstances described (essentially of him refusing to perform his marital obligations to his wife) I think any Beis Din will rule he must give it.