I doubt it. She might just think the neighbor got it by mistake.
The reason I posted, not like the others to dump it, is because you said that the store owner is frum. Perhaps his letters and calls aren’t being ignored on purpose. So by you putting it the mail box or under the door, you might just help him get his money, if your neighbor isn’t aware of the situation. If your neighbor is ignoring the store on purpose -they will ignore this card also. But by you leaving it, it will almost for sure advise them of the situation, if they really want to correct it, but are not informed.