[NYS] Who owns my cats?

My family has been taking care of 3 cats for nearly 5 years, a mother and her two children, the children were born in our house and have lived here their whole lives. The cats are chipped under a small local non-profit that my mother volunteered for until about a year ago, until she got into a disagreement that ended poorly with the owner of the non-profit 'Jane'.

One of them went missing a few days ago, and Jane was notified when the chip was scanned at a vet. Jane was told that the person who found him was in possession of him for a month, and she feels we aren't fit caretakers. We have photos and testimony that show the cat was in our possession until around Monday, when he most likely got out of the house due to contractors going in and out.

She refuses to return the cat stating to the effect she owns them, and is now insisting on picking up the other two. I suspect that she probably does have some form of legal ownership due to the non-profit's involvement in the spay/neuter of the kittens. Unfortunately, we never officially adopted the cats. We were initially temporarily fostering them, but she failed to find a permanent foster home for these three, and in that time we have been the sole caretakers. Would it be sensible to argue in small claims that we are the de-facto owners? The non-profit does not pay for anything like food or litter, and we have not been compensated. Jane has not been in contact with us at all for about a year.

We are making every effort to work this out without involving law enforcement, but she's refusing to talk to me, and is giving my mother limited information. I want to pursue the return of my cat, he means a lot to me.

Thank you in advance r/legaladvice. Let me know if I can clarify any relevant details.