Oh, the joys of living near people who want you to be just like them...
Since you're stuck with staying where you are right now, you need to get proactive. And since busybodies are found in remote rural areas as well as in town, any strategies you develop and implement now will serve you in good stead later.
I concur with the idea of pestering the code folks with polite pleas for clarification. They have to respond to complaints, but that doesn't necessarily mean that they are in sympathy with the complainants. At the least, I would request copies of the regs that outline which structures require a permit and which don't. And then call them regularly, asking if this is alright, or if that is alright...You want them to be able to say to any complainers "yes, we know about that, and Mr. Lynx is in compliance."
Are there other permies in your town/county? Find them, and see about getting a backyard chicken
ordinance. Or see whether 2 or 3 hens are already permitted as pets. If there are no permies, see about joining a gardeners group, and be active enough that they know who you are. If you find their meetings boring - and you may - volunteer
yourself as a presenter on a topic that interests you.
Do you share what you grow with your neighbors? Food, flowers, herbs - this can be a good investment on your part. If your immediate neighbors are benefiting from what you do, they will be likely to at least remain neutral.
Does the code allow small signs in your front yard
? Consider posting a sign that says "this yard produces X pounds of fresh food a year" or "wild bird haven" or something else that conveys the worth of what you are doing. You will probably never win over the original complainer, but others in your neighborhood could well be curious about what the heck you're up to. Let them know.
Above all, make sure that you are scrupulous about keeping the visible parts of your yard neat. No tools lying around, plenty of flowers, no dead tomato
Hope this gets you thinking!