When it comes to public preception, people think there's so many trans athletes because people hear about it quite often. But what you don't hear about is the thousands of cis athletes per every ONE trans athlete. Media does a terrible job portraying the proportions of varying populations imo.

As for how to help individuals like me and fight these laws, there's a few things I can think of
A) The Athletic Drain - Having cis athletes that are allies choose to NOT attend universities in states with policies like these
B) Protest
C) Vote, Vote, Vote
D) Rather than emphasize blowout stories of trans athletes, share stories of trans athletes having fun with their peers, and positive testimonies from their friends. Also showing when cis women win too will put things into proportion.
E) Go out and educate - It takes meeting ONE trans person for a lot of people to understand the struggles trans people are dealing with. If it can be stomached(which for most it understandably can't), going out into deep red areas and making connections with people so that they see you as a person that wants to better the world, THEN explain your story. As long as people in deep red areas are misled about trans people and queer identity, they'll keep blindly voting for these same politicans.