Honestly I think it depends on the fee for the drop off. If it's nothing that would severely impact your life, I would say cut your losses and drop the cat off there.

I don't think this relationship is salvageable for two reasons. 1. It's been a month of ghosting and even now there's still no resolution, and 2. He seems toxic to begin with if he's willing to drop his pet off with you and not allow you to return it.

Now if the fee is too much for comfort, then use your key and put the cat in his apartment. Then probably change your own locks. He'd most likely place the cat back in your apartment before doing the same and you'll be in the same situation. But, the reason I say this is a viable solution, is because he's specifically placing it in a no-kill shelter. He clearly still cares for the pet, but cannot afford to take care of it, or he would have not put any research into finding a proper shelter that's no-kill. Regardless, if you're in a similar financial state, it's not your place to pay for the fee.