As cathartic as shooting is, you're opening yourself up to a lawsuit even if they are on your property when you shoot. If they aren't on your property when you shoot, you'll lose that lawsuit, especially if they have a letter from you with written intent.

If you know which neighbor it is, they may need a face to face meeting. Bring up the goose and maybe the chickens. Definitely make up a story about coyotes or foxes and say something like "with all these predators around, I'm going to need to put out traps to protect my livestock, I'd hate for your dog to get its leg broken."