Petitions are good if you can get signatures, also getting the word out to others by newspaper, tv, radio, or simply person to person conversation is the best thing to happen, specify the issue at hand, explain the reason why the law is total bs, and point out exactly which town board members are the a-holes who are against a person keeping a couple birds, it is quite ridiculous, especially nowadays the trend is to allow urban chickens in regulated numbers.
This 250 foot rule is what tells the issue here, they made that rule because they feel chickens (which are livestock to them not pets, no matter how you keep them) are stinky nasty creatures and they want them far away from neighbors who would be offended. It needs to be pointed out that this is bogus as a proper coop with 2 or 4 or even 6 or 8 hens is not going to stink if properly kept up. A new law allowing chickens could still be quite restrictive in the towns favor, limit of 6 birds, no roosters, must be 15 feet from lot line, must stay on your property, no offensive odors, all these conditions could be kept in place and folks could easily have their small flock within that constraint. If this is pointed out there is no reason anyone should have an issue with it, neighbors will still be able to report problems if these stipulations aren't followed, any town board member who has an issue with that needs to be removed from office.
If you are not taking part in commercial farming or running a for profit hobby farm/petting zoo on your property I do not see how they can classify 3 pet hens as livestock, is a person who keeps a dog outdoors keeping livestock? This is essentially no different than someone putting a dog out in a kennel and keeping them that way.