And Tadano's crossdressing is EXTREMELY convincing. We know this now.
... Does he change his way of speech as well? How would they not recognize his voice?
I bet the joke would only work on manga form. If it's in anime form there'd be voices which would give away Tadano easily (except if they gave the girl version an appropriate falsetto).
Tadano: Good morning!
Anchi: And who the hell are you!?
Tadano: Oh right, sorry [puts on wig]
Tadano-kun-chan: <falsetto> Good morning everyone!</falsetto>
Tsuzurafuji: Oh hi tadano-chan!
Anchi: WHAT THE HELL?!