You could tie it in with letting the user specify the name to display; changing away from using Twitter handles is probably not a bad thing anyway. Thus you could have as part of the sign-up process:
1. Call me _________
2. Refer to me as
( ) male ("he")
( ) female ("she")
(x) gender-neutral or plural ("they")
( ) actually, we're a group ("they")
For e.g. Twitter sign-up, you'd auto-fill the first with their username and the second with gender-neutral.
That does sound rather awkward and could hopefully be improved upon. (I added the final option as it occurred to me businesses could use this.)
I agree that the concept of displaying "he" or "she" would indeed be an improvement, but I feel as though the only way to get over the awkwardness of the form questions you defined would be to go through some kind of "profile builder".
This obviously takes the app to a new place - now talking about some sort of profile system - this would obviously be a huge feature to add with a load of other implications.
That does sound rather awkward and could hopefully be improved upon. (I added the final option as it occurred to me businesses could use this.)