This is a very painful situation.
I think you need to not focus on what you will or won't do (break up or not) and instead focus on helping your daughter to speak about her feelings toward you, including how she feels about your involvement with this man.
Oftentimes when a person can be helped to talk about his/her feelings, the problem resolves itself, which means that you won't need to make a choice.
What we need to do is to help your daughter to become conscious of the feelings behind her ultimatum.
The effect of her demand is that she is forcing you to choose, and I wonder why. Does she feel that you don't have enough love to go around? Does she want to keep you all for herself? How is her behavior and the struggle you are in similar to what you experienced together when she was a child? Did she feel dropped or cast aside by you when she was young? Did she feel that you didn't have enough time for her?
Did another child come along before she was two years old? If so, she may have felt that she didn't have you all to herself for long enough before you turned your attention to another. (If this is so, your involvement with this other man may have awakened this early wound.)
Is she being loyal to your ex. (her father) by not wanting you to have another man? Are you experiencing a delayed fall-out of feelings of resentment from the time that you divorced her father?
Basically, what I am suggesting is that you explore her feelings and their origins with her.
Don't pass judgment. Don't suggest that her feelings are wrong (feelings aren't wrong or right; they simply are what they are and need to be respected).
If you find this hard to do, then go to a family therapist who can help you navigate the discussion. If you can get her to identify where her feelings are coming from, and how they relate to past grievances that she never shared with you, you have a good shot at resolving the real issue.
When she feels heard and understood by you, she will feel filled up and less in need of demanding that you have no other loves but her. Let me know how you make out.