I always suspected that the victim Rod Steiger picked up when he was in drag was ALSO a drag queen. IMDB confirms that the "woman" was in fact a man in drag.
I wonder if this was a casting choice so that Steiger would be more believable as a woman. If not, was the murderer aware that he killed a man, not a woman?
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Since the issue of the hooker's true gender was never addressed, I always assumed the filmmakers had simply used the biggest "actress" (in this case, a king-size female impersonator) they could find to play the hooker just so that, by comparison, the not-exactly-petite Rod Steiger would even seem remotely credible as a frightened little matron.(Sequence wouldn't have made much sense if the hulking Steiger-in-drag was being comforted by a normal-statured hooker much smaller than himself.)
For what it's worth, a 1968 issue of an expose men's mag called Man-to-Man contains an article about a New York City drag revue (google "Club 82") in which it's stated August had just completed a role in No Way To Treat A Lady where she played a "femme prostitute"--no reference to portraying a cross-dressing character.
A slimmer Kim August can be see in this 1965 program from Manhattan's Club 82.
http://www.queermusicheritage.com/fem-cl82b.html