Small town where everyone knows everyone. There's that ONE lady in town who makes shirts out of her house for her students and random things here and there and she's a teacher at school. Well now she must be making banners too. I had 3 phone calls asking for logos for a dance team banner. I'm hesitant to be rude about it and not send them. The customers all paid for their artwork but if I'm not making the banner then why should I be supplying the art for it. The problem is if I say no then she goes back and tells the customer and they get mad a ME since they paid for the artwork. The other hand is that's what sucks about doing sponsor banners is you don't have all the logos and if the customer doesn't give them to you then you have to vector them or reset them - like the rest of us have to do. GRRR...maybe I'm just being crabby about it...mostly the woman thinks her crap doesn't stink so I have an attitude against her already.
If this is the person doing the banner, I would tell her to have your customer contact you for the logo. Don't just hand out logos to people that other's have paid for.
Now, the logistics of if your customer is even owed the design, to me, it's about being pragmatic. If there isn't a contract that stipulates to such, that's on you. There should be a contract that stipulates that all artwork belongs to you or that you send the artwork to the customer after payment etc. It's all about not leaving a bad taste in one's mouth. Especially in a small town. Even if they are wrong about getting the design when there is no contract saying one way or the other. Word does tend to get around. At that point, it's going to be a battle of the reputations.
I, however, would not be passive aggressive and muck around with the logo etc. I have had some people do that before and when I get the file, I let the customer know (and I can tell based on the original finished product and the file that I am given that it has been mucked around, it is painfully obvious even to the customer when I show them how the new machine file looks compared to their product) and you run that risk that that too makes the rounds in a small town.
Other suggestions (one actually specifically in particular if you keep in mind that this is coming from me), I for sure wouldn't say, even though I get the joke, there are a couple of big tells for those that even just stop and think about it, they will soon know that you were shining them on. Which again, gets around in a small town.
In short, I personally would stay away with passive aggressive solutions.
Sometimes, no matter what you do, can cause a stink, but it all boils down to how much blow back you want to deal with.