I second the informal (and private!) discussion with the supervisor-provided you know your super won't do something amazingly dumb about it, like assign you two to work together to force you to work it out or some nonsense.
But right now? she's being a real creep, but she's not doing anything improper. And I suspect she's being careful to *not* do anything improper, so as to keep her job. I don't blame her on that.
I think your best bet is to leave her alone as much as possible. Beyond that, since she's obviously transphobic, I would suggest that she's capable of possibly making wild accusations in your general direction due to misunderstandings. Not likely, but I would avoid her right back, try not to notice her, look at her, or be where she is if you can help it without disobeying orders or peeing on yourself.
If you do have to talk to her, be very polite, formal, state things precisely and concisely, and then go away again.
Your goal here is to keep making a paycheck, right? not to get everyone to like you and get the whole world to hold hands and sing Kumbaya and whatnot.
I think it's good that she's the only one at this job with this attitude, also that she's not actively plotting a way to get you fired. I wouldn't rock that boat if you can help it.
And don't let her get you down either, my friend. You're good people, and she's a bigot. Haters gonna hate.