Here's the Fairness For All Act on women's sex-segregated space.

tl;dr Like the Democrats' Equality Act, the Republicans' Fairness For All Act allows any man to self-identify and access women's sex-segregated spaces (well, at least bathrooms and locker rooms)

The most telling thing about right wing criticism of the Fairness For ALL Act is that none that I've read mentions that it puts women's privacy dignity and safety at risk.

NEITHER PARTY GIVES A SHIT ABOUT WOMEN.

"Transgender men and women use restrooms and locker rooms for the same reasons everyone else does. And when they do, they value safety and privacy just like everyone else.

It’s already a crime to enter a restroom or a locker room to harm someone. With or without this bill, illegal behavior is illegal.

Protections for transgender persons like those found in the Fairness for All Act work.
For years, over 20 states and more than 200 municipalities already have laws protecting transgender people from discrimination.

The FFA Act explicitly prohibits people from asserting gender identity for any “improper purpose.”

Nothing in this law weakens existing laws against illegal behavior. Assault and harassment remain illegal.

The FFA Act expands privacy by requiring public schools, public accommodations, and employers to accommodate requests for additional privacy. This will increase the availability of private, single-user facilities for everyone.

Provisions like those in the FFA Act have widespread support from hundreds of organizations across the country.

More than 250 organizations that do work to prevent and treat survivors of violence support laws that protect transgender people in restrooms and locker rooms.
These include The National Alliance to End Sexual Violence, The National Center for Victims of Crime, The National Coalition Against Domestic Violence, and more.

https://fairnessforall.org/resources/