I'm in a similar boat, but I found ways to make faux-extroversion work for me.

What I basically found was that if I engage with individuals at group events as though we were alone, then all I have to "fake" is the being-in-public body language... basically, I have to have individual conversations as though I were on stage. The emotional impact is similar to that of a private conversation, and having superficially public conversations that have the internal structure of private ones seems to really convey extroversion, since I'm talking about stuff people don't tend to talk about in public.

But it takes a certain amount of self-monitoring and steering to remain in that state; once I tire out I can't maintain it anymore.