Since this is a small scale local type of law, I would try to get face-to-face meetings with the people on the city council (or whatever applies). Talk to them about how reptiles are becoming more and more popular as pets, and that most reptile species are small and harmless.

It seems likely to me that this bill is really aimed at animals that ARE inappropriate in city residential areas, because it starts with "livestock". It might be very easy to get it re-worded to specify only reptiles over a certain size and/or venomous ones. Since this law only requires permits, and is not an actual ban, I think that would be reasonable.

I would also recommend trying to get answers on how easy it will be to get the permits. For this, you should probably talk to the commissioner of public health, being that they are the one who would have to give the permit. Some "permit required" things are really just ways to generate revenue. You fill out a form, pay a fee, and approval is automatic or close to it. In other cases, permits will rarely if ever actually be granted.

You should also look in to what pet stores in the area would be affected by the new law, and try to get them involved. Even if you don't get cooperation from the stores, it is still something you can bring to the attention of the city council. Law makers on any level do not want to hurt the economy, although it often seems they are too stupid to realize that the new laws they propose will have that affect.