Hi there-
As Miss Prissy said, we did go through this with our town last year.
Our code stated "no livestock" as well. In the "definitions" section of the code (at the start of the section), livestock was defined to include poultry. First off, I wouldn't wait till they upload to municode. Go down to the town hall and READ the code. They have to have one someplace that you can look at. Find the definitions section at the beginning of the section that says "no livestock"... it will probably be EITHER in the Zoning/Land Use section (for your particular residential district) OR in a special section like "Animals and Fowl" or something like that. Read that definition of livestock. if it doesn't specifically say poultry, fowl, chickens..... you are fine IMHO.
OK, if it DOES classify hens as livestock...... there are a bunch of things you can do, on a scale of drastic-ness from quietly negotiating with the code enforcement officer...... to..... starting an all-out campaign to change the zoning ordinance. The latter is what we had to do, and it was a LOT of work and time. I would be more than happy to talk to you about it and share the resources we collected during our summerlong campaign last year.
Before you try to negotiate with the city, I'd collect a bunch of supporting evidence (I can share with you some, if you want...email me). First off, you need to make it clear to them that you are only keeping 3 HENS (no roosters) as PETS (not barnyard animals). I can just imagine that one of your kids told a friend you have 12 chickens and the parents were like, "Not in our neighborhood!!"..... I can tell you from experience that a LOT of people have very strong negative preconceived notions of chickens as stinky, dirty, noisy, mean, farm animals and associate them with a sort of backwards, trailer-trash thing that they DO NOT want in their tidy suburban neighborhoods. 
Our eventual success was because we really hammered home that the "pet hen" movement (we changed the wording from "backyard chickens" that we started with, because we needed to stress the "PET" angle) is a rapidly-growing trend in "MOSTLY UPSCALE NEIGHBORHOODS" and that it's mainly the green-conscious, educated, higher-income Whole-Foods-Market shoppers who are doing this.... and then to really convince them that your 3 birds are PETS and should be regulated (if at all) just like any other suburban pet (i.e. dog, cat, rabbit) and that they will be cared for by you just like any other suburban pet. In other words, you're going to clean the coop every other day (or whatever) and that your 3 hens will produce less poop that a small dog..... I have a whole list of rebuttals for all the opposition arguments we encountered, if you like.
If the code officer is driving around your neighborhood, I'd get on this ASAP... get that livestock definition so you know what the law really says, and get your (polite, respectful) argument all organized with supporting documents, etc.. and then talk to that code officer right away before they can get much further with their investigation.
If you get nowhere with that, the next step I would take would be to find out what your town's procedure is for getting a variance. Usually you have to make a presentation to the city council and/or planning board. Again, I have lots info if you decide to do that. Barring a variance, the only thing to do is try to get the law changed. Be prepared for it to take a really looooong time (we started in May and it didn't get resolved till late September) and hundreds of hours of your time.
Best wishes,
Stacey