No 'gender cops' required, as gender is irrelevant. Sex is usually obvious, & more importantly, most people are happy to abide by nice clear rules, like using the toilets EITHER designated for their own sex OR if for whatever reason (personal identity, caring for an opposite sex adult or child) to use appropriate mixed sex facilities.
& where such useful mixed sex spaces don't exist, we can all agree they should be lobbied for, to go alongside the single sex provision that is already there.
This isn't as difficult as you seem to want to make it, jj.