My headcanon is that Lebanne sought Jacinthe out for the express purpose of getting Jacinthe to pay for her top/bottom surgery and hormones. She knew Jacinthe was rich and heard that Jacinthe made bets, and was confident enough she could win that she went for it.
So she challenged Jacinthe to a battle, and the conversation went like:
Lebanne: "Jacinthe! Battle me!"
Jacinthe: "You wish to battle moi? How intriguing! Pray, why is it you wish to battle?"
Lebanne: "I wanna make a bet. If I win, you pay for my top surgery."
Jacinthe: "Alors! Very intriguing indeed. And if I win?"
Lebanne: "I'd do whatever you want. Not that you stand a chance."
Jacinthe: "Very well. I accept your terms. May I have your name before we start?"
Lebanne: "[Dead Name Redacted]"
Jacinthe: "Thank you. Now let us begin!"
So they battle, and of course, Jacinthe wins. Lebanne is gutted for more than one reason, but is a woman of her word and so doesn't try to back out. Jacinthe says that Lebanne is to be her maid, but also that, after they get her fitted for her new uniform, they will need to look into scheduling her surgeries.
Lebanne: "What are you talking about? I lost."
Jacinthe: "I don't recall saying I wouldn't pay if you lost. Why, this is about your health! Scheduling these procedures is of the utmost importance."
Lebanne is flabbergasted, but she's not about to argue when she can't afford the procedures herself.
One final note is that when Lebanne is next asked what her name is (before the procedure), she starts to introduce herself by her dead name again, only for Jacinthe to put a finger to her lips.
Jacinthe: "Ah-ah! You gave that name to moi, don't you recall? That name belongs to me now. You are not permitted to use it. I recommend you choose a new one."
And Lebanne is once again bewildered, but if she's finally going to present as a woman anyway . . .
And that's when and why she names herself Lebanne.