In that case, the cross-dressing is a means-to-an-ends more than (or instead of) a reflection of identity. Unfortunately, there is no quick way for someone to prove they are the latter rather than the former (short of ID). But at the same time, a "man" doesn't need special clothes to sneak into a women's restroom, and some adjustments can be made to ensure safety of people in a restroom much like in a stairway. Aside from discouraging sex-related behavior by cisgender heterosexuals, sex-segregation (versus gender-segregation or even no segregation) doesn't do a whole lot to ensure personal safety and prevent crime, so society would do well to think more deeply about the logistics involved.
Of course, when your society has lots of people who think TGism is immoral, it is difficult to work with them. The best you can hope to do is get enough clear-thinking individuals to assemble an effective policy with useful recommendations (to assist the transition). You won't satisfy the fringe, but you may weaken the appeal of their rhetoric and/or get most of them to grudgingly play along through the power of the comprehensiveness of your policy -- you don't give them what they ask for, but you address the fears they have (empowering and assisting them to ensure safety).
