Probably shouldn't have shot it but it's not that doing so is not within your rights. But the real asshole here is the neighbor. They are acting inconvenienced by the fact they won't take responsibility for their own cat and are now having to pay for damages it did. THEY should have been taken the prerogative as soon as he contacted them to offer to pay for whatever the damages are. I mean, if their kid had gone into his garage and destroyed $1400 of stuff, you think they'd act any different? Yes, it was unintentional but they are still at fault. Most people who aren't total fucking dickweeds that tend to push their responsibilities off on others would feel at a sense of duty to rectify the situation their dumbasses created.
When people are too lazy to take responsibility for their own shit, that is exactly when you invoke calling the police or whatever you have to do. Especially when you've asked them before and they don't do dick about it. We don't (as a society) condone vigilantism and therefore, you are supposed to call the police. Would they rather have you just killed it?
To do otherwise and just ignore it would be being a pussy and pushover and letting them bully you through complacency and inaction.