Speaking from the standpoint of a town official, I'm afraid you have put yourself in a difficult position. You'll be entering talks with your officials from the standpoint of someone who has already shown that rules and regulations aren't important. So what are you going to be able to say to convince them that now you are willing to follow the rules - as long as they let you keep your animals. Before you get defensive and start calling me names like Nazi, Gestapo, or any of the other things I've been called when I bring up these points, think it through. You had chickens, which weren't allowed where you lived, but you did it anyway and had no contact with the officials of your town until they came to you. Your chickens and rabbits may be kept in perfectly clean, adequate surroundings, but they are still not permitted. They may be quiet and peaceful and unobtrusive, but they are still not permitted. You may have had the best intentions of trying to be more self-sufficient, but they are still not permitted. So when you meet with the town, the first thing you'll have to do is convince them that you didn't mean to violate any ordinances, but their point will be that you did exactly that.
So your job is two-fold. First, you have to convince them that you had a good reason to bypass seeking a variance before you put so much hard work and money into your project. That's going to be tough. Then you have to prove to them that your critters are housed in a sanitary manner, healthy, and are not annoying to anyone living around you. Again, that might be tough, because obviously someone filed a complaint. Most of us who serve our communities also have full time jobs outside of town hall and are not in the habit of getting into our cars and cruising neighborhoods hoping to find violations of some sort. We usually don't know about an issue until someone brings it to our attention - and that means that someone turned you in. I've also been told here in these forums that having chickens is no big deal and that greedy town officials could have chosen not to act on a complaint. My response is always, "Really? So if you went to your town government with an issue you wouldn't mind if they decided your problem wasn't a big deal and ignored it?"
You're going to need letters from your neighbors stating that they don't mind one bit that you have chickens in your yard. You're also going to have to provide some kind of veterinary or state inspection certifying that your animals are healthy, that they pose no disease threat, that your property is free of feces, odors and other nuisances. Photographs and a invitation to visit your property personally are good suggestions. And finally you're going to have to do all of this in a calm and logical manner. You are going to need to be pleasant, cooperative and persuasive. I don't think lying to them about your chickens and rabbits being "service animals" for some ailment or something is going to get you very far, and that has been used to the point where it's become difficult for people who really do have "service animals" to be believed.
I'm not trying to criticize you personally. I served on my town's planning and zoning commission and currently sit on our town council and I do own chickens in town. I happen to be fortunate enough to live in a community where even in town folks have horses, mules, donkeys, steers, lambs, goats, chickens and roosters on their property. But we have ordinances protecting the non-animal owners in town from nuisances and we have a plan in place to deal with complaints plus a permit system. When you appear before your officials, remember that you are not only trying to get permission via a variance to keep your chickens and rabbits, you are paving the way for future backyard chicken owners. I wish you all the luck in the world. I would be devastated if I had to get rid of my girls, but I also know that I have a responsibility to the majority of folks in my community to enforce the ordinances on the books fairly.
and Welcome To BYC! Sorry about the problems you are having, be sure to check out the Laws forum, the sticky threads especially have links to help with changing ordinances etc 
