I offered to pay for a vet to put to sleep my neighbor's annoying dog
You can't make these kinds of decisions about your neighbors animals!
I know it's not polite, but her dog is too noisy. But if I put my neighbor's dog to sleep, she'll sue me for murder
Exactly! Maybe talk to her about the barking instead! It is not okay to tell your neighbor what to do with their animals.
But she's a bitch who doesn't listen to me. I can't live with a dog barking all day!
Have you tried talking with your neighbor about it instead of telling her what to do? They will be more willing to work with you that way.