It’s not a bad idea. I put up a chicken wire fence inside a 4’ chain link fence when I had two overzealous Rhodesian Ridegebacks running around always acting like a-holes right at the fence line. Basically, a fence within a fence.
You do you, but i would suggest you don't take it out on the dog. Dogs are going to be dogs. It isn't their fault their owner is a deadbeat.
They only care if it is visible. They could care less about side fencing that is not visible from the street. HOA are pieces of ****.
I had a similar suggestion -- build your own fence on your side and keep it maintained. It's an expensive solution for someone else's negligence but it'll get the job done. If they can break a wooden fence, do a hurricane fence. Chicken wire is a much cheaper solution, but might not be strong enough.