Similar to what we have here. My dogs have killed on and mauled another of a particular neighbors dogs. Those neighbors re-dog quickly and they have lost multiple I cannot account for.
For fun, could you post a copy of the HOA rules and indicate the county and state in which you reside. it might help us all understand better how the rules mesh across jurisdictions.
Put hotwire directly on coop / pen area so dog gets zapped when contacting it. It is likely that would not qualify as a fence and would not be at property border where it would be seen. If a dog owner complains, then you would have their admission the dog is on your property setting up so HOA...