Hmm... sounds like one for the lawyers, which I am not, thankfully. Their performance. Your recording. Somebody else's music. Quite a mess you have there, even without dodging down the copyright rabbit hole. Your claim is you own the recording because you recorded it. They can claim they own the performance, without which your recording would not exist, so they own the recording as well. It sounds to me like they have the prior claim, but then what do I know? Not much...
I would give them the recording with limited rights for personal listening, excluding such things as using the recording to solicit new gigs or selling the recording. If they want to do that, then go for the money if you want, but you are still in a very weak position, rights-wise, I think. So build good will and suggest that in the future you would consider producing a recording for them they could distribute, and get a contract beforehand. It saves a lot of trouble.
We all want to be compensated for our work, but sometimes it makes more sense to give it away in hope of future paying work. Sometimes it's called advertising.
(Last time I looked at a cover jacket, it was pretty clear the work was owned by the performers and their producer, not the recording engineer. But then what do I know?)