What are the options?

1) Gender-neutral facilities: this means that North Americans need to be more like Europeans and get over their Victorian-era inhibitions (only slightly more likely than Americans giving up their guns, at least without significant 'social engineering', but which I still see as a desirable end goal).
2) Use gender presentation and/or gender identity as a basis for who can use which washroom. This means that someone who seems to be of the 'wrong' gender will occasionally pass through those doors, and requires that we trust people to not engage in untoward conduct (which currently governs our bathroom behaviour, by the way).
3) Status quo. Does not eliminate ambiguity, as we have seen in the cases of people with non-stereoypical presentation, e.g. butch women.
4) Require ID and 'potty police'. Same problems as #2, depending on (inconsistent) state rules for changing gender markers. Even federal standards on who can claim which gender (which states may not accept) still fail at least some of the time because not everybody fits gender stereotypes (see #3).

The issue is a balance between current (North American) social mores and perceptions of harm vs tolerance of non-stereotypical expression/breaking down gender stereotypes. For this to advance at all, we need to 1) change the current social mores as they are embodied in an understanding of gender, 2) educate people on altering risk perceptions to match fact, and 3) work on reducing sexual violence of all types. It may take a generation or two.