I believe you when you say there aren't rats there. The actual pest here is the HOA and the neighbors.
Nevertheless, at some time in the past you chose to buy into a HOA-governed subdivision for a reason that appealed to you at the point of purchase. Perhaps you enjoyed the orderliness or wanted to be with similar young families raising children in the neighborhood....in other words, pets and chickens weren't in the picture when you bought into the HOA.
Years ago I lived in a vacation condo near a beach...and I didn't even have a cat much less a hen or heaven forbid a Rooster!!!
. The neighbors freaked out about stuff like SHOES left by the doorstep to air out. Moved out of there about 5 years before Hurricane Katrina blew it all away LOL. Got a cat pretty quickly after I moved out of the condo though.
You didn't have to buy big-HOA-government property but there you are....it might not be the Feds but you're nevertheless under a big HOA government of sorts.
So what to do??
What does your HOA allow? Does the HOA permit only cats and indoor pets like aquarium fish? Dogs (small only?)? Or better, dogs (up to 3 or so of any breed/any size?) Or even better, 1-2 Horses? Your chances are best if either a horse or any size of dog is already permitted. If they only permit indoor fish and a couple cats and one or two under-25-lb (small) dogs I strongly doubt they will ever allow hens.
Petition through appropriate channels for rule changes or else go somewhere that respects your freedom more and is more chicken-friendly.
Do EVERYTHING you can to keep any trash appropriately picked up and disposed. Usually excess trash on a property has NOTHING to do with chickens. I figure the chooks are happier with trash cleaned up unless it's a weekly-maintained compost heap that creates treats for them.