Whenever the spurious "racially segregated toilets" analogy comes up, I find it useful to compare and contrast the reasons given for the segregation in both cases.
Racial segregation of toilets in the American South prior to the Civil Rights Movement - the reason came down to this: whites thought blacks were lesser, and whites held the power.
Sex segregation of toilets in pretty much every country in the world (barring a few western nations which are now trying to make a gender neutral point) - borne out of the fact that men commit 99% of sexual offences, their victims are predominantly women, and that without single sex toilets women are unsafe (see, for instance massive prevalence of rape in rural India in areas without indoor toilets where women have to go into nearby fields to relieve themselves).
So in one instance, the imposition of segregation is driven by arbitrary prejudice on the part of the dominant power grouping in a society, in the other it's driven by a desire to protect the less powerful group from a real, and well-documented danger from the more powerful group (NAMALT etc. but enough of them and not conveniently marked with the mark of Cain upon their foreheads).
So sex segregation is completely different to the Deep south "color bar."
The question then becomes whether TWAW and should be included in the social category of "people allowed to use the women's toilets" (i.e. that category should be determined by something other than sex). The existence of individuals as clearly dangerous as the person in the OP makes me think the answer is a clear no.