I had three feral cats hunting chipmunks, lizzards, and songbirds, and chasing turkeys on my property, and a rifle. I still have the rifle. A cat that is not in it's owners house or yard is a feral cat.
In your situation, I would:
1. Try to talk to the neighbor. He may understand. Many don't.
2. Use a deterrent, such as a dog urinating there and chasing the cat.
3. Capture the cat in a box trap baited with tuna, and take it to an animal shelter. You can always say that you did not know whose cat it was.
I would not cause pain to the cat. You either kill it, or you don't. Also, ask yourself how would this change your relationship with your neighbor.