I would tell the neighbors that you are afraid for the cat if your dogs get it. "They have killed before and might kill again." Then let the neighbors solve the problem for you.
I used to have cats, but somehow started getting allergic to their hairs, or whatever. Here it is the difference that I see now, between Cats & Dogs:
* Cats think they’re Kings and you ought to worship their stinky arses. When you go to the grocery store and comes back, they will be laying over the couch and will turn their face to the front door after you get in and their face will be like “oh, it’s you again... never mind! Please bow down and worship me!”
* Dogs, they think you are some sort of a god and need to be worshipped and protected. You go get the mail outside, as you come back in they will be celebrating in happiness to see you’re back in again, and they’re like “oh, it’s you, my good Sir. I am so happy to see you’re back... What can I do for you? Can I wag my tail a bit faster for you? Can I kiss you all over? Damn, I love this guy so much!”
Owners are responsible for their pets. When they start prowling around my house, laying on my cars, truck, patio furniture etc. it’s my responsibility to handle it. I wouldn’t ask the neighbors ****, then they know your not happy, the cat goes missing, your automatically the suspect. Just handle your situation.