I don't know. If you end up having her stay, you can just say that she didn't fight, I guess. I mean, she and Mai didn't seem to be doing anything during the eclipse invasion, so it's probably safe to say that they don't accompany Azula *all* the time... maybe all they did was help track down the Avatar, and when they were at home in the Fire Nation the fighting was just left to Azula and her Doom Army. *lawl, total guessing here*

But if you do want Ty Lee to leave, I don't know how to involve Mai... because while Azula might eventually start thinking of her as a friend again, I don't really see her as trusting enough to let her in on her plans and stuff, and she'd- at least in the back of her mind- always be expecting a betrayal... so at least the Mai!canon of TBR wouldn't go down.