This is a link to the HERO ordinance that was adopted by the Houston city council in May 2014. The recent election was about repealing the ordinance. Houstonians voted 62% to repeal.
https://archive.org/stream/equal_rights_ordinance/equal_rights_ordinance_djvu.txtThis is a long ordinance. It classified who civil rights applied to, including gender identification. This ordinance went past protections from federal and state laws. A city can extend rights to additional person under the concept of home rule. This is done by action of the governing body or electorate action. What Houston did was fine.
The electorate decided to vote to repeal the ordinance. I concede the measure on the ballot was likely short, but much information was in the public square about what was in the ordinance. The ordinance has a section on discrimination and denying public access to public accommodations.
A bathroom is a public accommodation. Opposition used a practice of liberal progressives who use vagueness of a law to get what they want. Progressive use legal actions. The opponents used hyperbole to push for the repeal of this ordinance. They realize that liberal progressives will take such laws and use them for the basis of attacking traditional norms.
Houston or even Renoland can pass such local ordinance, and extend rights as they so define. So unless a court establishes a right for gender identification, voting is the only way to establish such a local law. Many cities have done so. Anyone who wants such an ordinance should have the fortitude to explain why an ordinance is needed and convince others. Same sex marriage proponents had a simple argument that has a majority support now. I do not support gender identification as a civil right, but would except it if it is the rule of law.