Don't relocate your chickens.
They are going after you for practicing Animal Husbandry--in a very small way you are--about the same amount as someone with a small vegetable garden is farming. In a very general sense they are not wrong, but the scope is very different.
Look through the Agricultural Section and find things that reference the scope that they are outlawing. I believe that you will find that its purpose it to prevent large commercial endeavours in a residential neighborhood.
In preparation, define exactly what your chickens are for your family. As much as possible correlate that to more generally accepted pets/hobbies.
Take photos of your family members interacting with the chickens. Have any appropriately aged kids join 4H or FFA, and provide some of that information.
With luck the city council will back down at your next meeting, but don't count on it, especially if there are neighbors in opposition to your birds. The fact that the City Council allowed your neighbors to prevent you from speaking strikes me as a City Council who is not opperating in a professional unbiased manner. Either the person chairing the meeting cannot maintain control or chose to not keep order. With a lawyer you may get the respect and hearing you deserve, but you may well have to sue in court. You will certainly have to sue to get your expenses reimbursed, although the notice of your intent to sue may cause them to rethink their position.