If we have to prepare for Mutually-Assured Destruction, so be it. Start researching the HRC, its members and their backgrounds. I'm sure you'll find lots of hate emanating from their mouths over the years. Start researching the activities of groups the HRC has aligned themselves with and supported. Because there's no way you're going to be able to frame this as merely the Fresno FRN chapter vs the HRC if it gets more news media attention.
The optimal thing will probably be to just support the mayor. If he can get the HRC suspended or shut down, they lose their legitimacy immediately (what little they ever had) and nothing Reyes or any of the rest of them have to say will matter. All we'd have to do is say "She has been completely discredited. We thus couldn't care less what she has to say."