Granted, I don't have a lot of experience with this issue, but I do have some. Most of my "out in public" time has been in Las Vegas where I try to use the bathroom in my room before going out, family bathrooms where I can find them or the large ladies rooms near the convention floor that normally are well away from crowds. The times I've had to use other, smaller restrooms have all gone on without comment, and I probably attract attention. On one occasion I was using the ladies room in a resturant and while in one of three stalls I heard a woman bring her young daughter in. I just stayed in my stall until they were done to avoid the situation, then as suggested took care of hand washing, hair and makeup checking as quickly as possible.
Based on my smaller sample and non-passing status, it may be a surprise that I've not attracted this question yet. I may just have to answer, "Yes, but if I were to use the men's room like this I'd get beat up." Then maybe a comment about how much nicer the ladies rooms are, depending on the response. I'll just have to see what happens if I ever face that situation.
In the meantime, it will be good to watch how those Bathroom Laws develop.