My mom taught me long ago that when people do mean, malicious things behind your back, the best thing to do is to talk to them face to face and apologize profusely for whatever you may have done. This usually works VERY well. You have to be completely sincere about it. Sarcasm ruins the whole point.
For example, I would walk up to the neighbor's house and knock on the door. (Do not be sarcastic in any way) When they answer, profusely apologize for your dog being off leash. "I am so sorry that _____ got away from me. I had no idea how scared you were of large dogs being off leash. I am just kicking myself in the butt for not being more careful and making you uncomfortable. I will be so careful with him and make sure he does not get off leash again. Is there anything I could do to make you feel better? Would you like to meet _____ and give him a treat so he does not scare you anymore? I know it must have felt awkward for you to tell me and that is why you called animal control. He told me you were so upset. Don't worry, after the 150$ fine I got, I will be very careful from now on"
Bullies hate being confronted and something like this would embarrass the crap out of them and make them feel guilty to have to admit they called AC on you. After they see how nice you are and having to see in person that you are genuinely concerned about their feelings too, maybe they will open up to you. If their cat is prowling the neighborhood, perhaps they were scared for their pet?
I have done this with real-life bullies in school, in sports, at work, etc, and I have ended up becoming good friends with all the people who were malicious towards me in the beginning.
I have learned that most people who do hateful things towards other people don't have much love in their lives and will respond very well to sincere kindness and caring. To "fight back" only excites them and allows them to continue their hateful actions.
I hope all goes well!