sorry, I feel for you, your neighbor is obviously coocoo for coco puffs, we had to shoot a cat a week or so ago, however it was definetly feral and in the chicken coop. (must have noticed the little peeps) I don't mind cats but letting them roam is too irresponsible and definetly a nusaince to everyone (enviroment included), not to mention the parasites it brings with it. I'm not talking about cute fat fluffy house cats, I mean minature panther types. If your neighbor cared about the cat they would keep it away from your place, I wonder if they are up to date on vacinations and all, doubt it. I'm thinking worms too....ug.
Unfortunately, it seems your only option is to very quietly trap and remove it, you prob won't have to lie about anything since they likely won't talk to you...please be careful about petting this thing - if it's anything like it's owner, you could get a nasty bite.
Why don't you shop on line for a better price on the traps, I know they are way overpriced, try hunting outfitters too, they carry diff sizes of traps and maybe you can reuse it for chicks or something in the future, domestic cats are usually pretty stupid about traps, so maybe it will be easy to lure.