Trans people will be murdered over bathroom laws. You want a masculine bearded guy in the women's room? That's exactly what anti-trans bathroom laws require. Those guys are gonna get shot.
Or, hear me out, we repeal the laws and trans people just continue using the bathroom of their choosing like we've been doing without issue for decades.
Remember in the 2000's before gay marriage was legal a lot of people would suggest elimination of the word marriage and just call everything civil unions to appease religious objections?
That's what eliminating gendered bathrooms sounds like. Gay and straight people enjoy being married. Trans and cis people enjoy their gendered privacy that makes them feel safer. I believe you're intentions are good but removing a universal cultural norm isn't the solution. It's side stepping the issue of transphobia, exactly like whites only bathrooms sidesteped racism.
To be honest, I don't want to minimize the problem or make excuses but I don't think trans men are going to be the targets of this. It is (trans) misogyny and going to get targeted to trans women nearly exclusively