Well, it looks like a few companies might have the right idea:
From:
https://www.bbc.com/news/entertainment-arts-48457401
Bob Iger, the CEO of Disney says Georgia's controversial new anti-abortion law would make it "difficult" for the company to keep filming there.
...
Earlier this week, Netflix said it would "rethink" its operations there with chief content officer Ted Sarandos explaining: "We have many women working on productions in Georgia, whose rights, along with millions of others, will be severely restricted by this law"
...
Comedy actress and writer Kristen Wiig also confirmed to CNN that her new comedy had pulled out of filming in the state while the executive producers of a new Amazon show The Power said it was no longer scouting for filming locations in Georgia because of the controversial bill.
Given the way that republicans have manage to stack the judiciary, I think economic pressure may be the best way to go. A lot of movies and TV shows are filmed in Georgia (thanks in part to government subsidies), but if production companies (and other businesses) start leaving the state, it may cause them to rethink their stance.
The same thing sort of happened in North Carolina... the state brought in a law regarding trans people and their use of the bathroom, which was widely criticized. Several businesses threatened to either pull out of the state, or stop planned expansion plans. As a result the state backed off its initial plans. (They still have some problems with their laws, but economic pressure did force them to back off the worst of their plans.)
https://www.usatoday.com/story/news...carolina-bathroom-bill-transgender/729791002/