I don't know if this will be at all helpful, but it's just one dumb thought that comes out of my mind, a cesspool of stupidity!
I personally despise HOA and the idiocy that stems from them. "Subdivision Nazi" is the term that comes to mind. I don't have one, my property has been in my family since they farmed it in the 1940's. Alas, I have been surrounded on all sides now by subdivisions. Several of the nazis at times have tried to impose their rules on me one way or another, I've told them to get lost and they went away, in light of the fact that I do NOT live in their subdivisions, the one behind me nor the one across the road from me.
I am sure that, in any sub, there are all kinds of rules broken. IF I were in your situation, I would do the following, but VERY QUIETLY:
Keep your chickens where no one will know about them. Make sure they can't be seen from the outside. Soundproof the coop as much as possible. Get a very quiet breed. NO ROOSTERS. And, try to get some breed that is very "exotic" looking == maybe some of the "show" type breeds with crests, fluffy feathers, that sort of thing, so they don't even look like what the average person would know to be a chicken. And, don't tell anyone they're chickens.
IF someone ever does see them who might make it an issue, make sure they know that 1) they're your pets: 2) DON'T tell them you eat the eggs, they might put 2 and 2 together; 3) given them some nonsense name that won't yield anything on a google search, and tell this person they're really rare exotic pet birds, make something up if you have to, call them "Indonesian Crested Sunbirds" or some other nonsense name.
Probably an UTTERLY stupid idea, but I thought I'd throw it out there.