This is all so depressingly familiar to me.
I grew up in South Africa during the Apartheid era; I was there during its fall and I remember the angst that this caused certain sub-sections of the white population, especially when public toilets became a hot-button topic. We used to have racially segregated toilets: there were white men's toilets & white women's toilets, and non-white men's toilets & non-white women's toilets. By law, you had to use the toilet that aligned with the race classification on your birth certificate and if a person of colour was found in one of the white facilities, they'd be arrested and imprisoned. People whose appearance was racially ambiguous were challenged by Security or the cops and they had to provide ID proving they were entitled to use the white facilities (obviously nobody cared about challenging people using the non-white facilities).
These laws were put in place to preserve the comfort of certain groups of white people who felt uncomfortable with doing anything intimate (like using a toilet or urinal) in the vicinity of someone who might be more talented at producing melanin. This segregation was scrapped with the end of Apartheid, and when that happened a whole bunch of people started bleating to the Media in a flat panic about what would happen to 'their' women when black women were allowed to go into the ladies' room with them. OMG, they might attack our women and children!!!1! They might molest them!! They'll kidnap our kids and chop them up for muti (traditional African medicine)!!!!

So you can imagine my exasperation when I see the same tired arguments being regurgitated yet again; it makes my eyes roll so hard that I feel like a human slot machine. The same tired rhetoric, the same lame arguments, the same mountains being made out of the same molehills. I hope these people are proud of themselves, because they're using the exact same arguments that were used to justify Apartheid.
You know what
really happened when the toilets were desegregated in South Africa? Nothing, of course. People kept doing their business, the sky didn't fall down, Armageddon didn't happen and within a few months everything had calmed back down. And that's exactly what's going to happen in this case, too. But it's going to take quite a bit longer for the tide to turn in the States, because (unfortunately) the Religious Right in America is very well-organised (and well funded!) in a way that those particular South Africans simply weren't, so they're able to mobilise huge numbers of people to spread panic amongst the ill-informed and to get actual legislation passed.
To counter this, you'll need to do two things: 1) use whatever political means you have at your disposal to work on repealing such legislation and to make it impossible for it to happen again; 2) take part in grass-roots activism to demonstrate to the rest of the country that trans people use the same public spaces as everyone else (and have done for decades without them realising) so there's no need to panic.
South African public toilets only became normalised when honest, ordinary, everyday people of colour simply started strolling into the 'white' toilets, went about their business whilst minding their own business, and strolled back out again. When nothing bad actually happened, people started to calm down. And that'll happen too when trans people do the same thing. It's clearly begun with some high-profile people taking a stand, but it's important to keep up the momentum and that means everyday trans people will have to do whatever they can too. But you need to keep fighting the good fight and never give in: after all, Apartheid was so well-entrenched and supported that it lasted 46 years.