"But this cat might belong to one of my neighbors"

I'm currently trapping in a neighborhood with a lot of ferals/strays. The woman who feeds there gave me a list of the ones she THINKS "don't belong to anybody." The association who is generously funding all the s/n, testing, and vaccinating is gently adamant with me that "no owned pets be trapped."

I'm in this awkward position of trying to judge cats in traps: "Do you belong to somebody?" I released one Persian I caught feeling confident this "talker" and striking cat was owned, but as he ran out of the trap I saw the horrible matts on his underbelly and brushed his emaciated ribs. If he is owned, those people are terrible to him. I regret releasing him and haven't been able to recatch.

I want to take in whatever goes in the trap but I don't want to cause community friction either. I've moved to a small town in the midwest, where TNR is "radical" and gets raised eyebrows from most.

What would you do?

I should add that "just ask the neighbors" isn't really a good option. Because TNR isn't exactly popular here, I think advertising what I'm doing isn't wise.

I honestly think that any cat that is able to reproduce and is let outside is fair game for TNR. The only question I'd have is whether to eartip.

Katie