Reads like she is uncomfortable about your behavior to the point of rejection if you don’t change your ways, even if you assumed her open-mindedness would accept what you told her. It's hindsight now, and no advice will undo you spilling the beans to her, so to speak.
Don’t have enough to go on with the dynamics of you and your daughter in limited writing, but going forward, try to be a little more aware by asking questions to get a sense of where someone is coming from, for instance, even if it’s family. Find out more about what your daughter accepts or doesn’t accept before you speak so freely with the women you keep company with. But since you feel “torn” between two desirable outcomes, and feel pressure from the ultimatum, let’s put it in perspective.
First, there is no one solution to solve your dilemma (as it hinges on an ultimatum generated by you daughter (which pertains to emotion), but think about loss in this sense for illustrative purposes; that is, if your daughter really likes the relationship she has with you, her ultimatum for you to choose between her and the young escort (who you may spend limited time with throughout the year), becomes the outcome where she will lose out to all the good you gave her as a father because of your one choice. On the flip side, you will lose out on your daughter’s acceptance because of what you revealed to her.
Thus, the above creates a double loss, and that’s always the big problem with ultimatums; nobody wins.
On the other hand, and this may come across as something spiritual or religious (but so what), it has its virtue towards peace of mind. I’m talking about forgiveness. It’s up to each to figure out how far to go with carrying guilt and resentment. Although, if there is any consolation, your daughter generated the ultimatum; so she has her work cut out towards forgiveness more than you, particularly if she doesn’t get her way and goes forward with her rejection. (She had other choices in how to go about it with her disapproval—why go through life carrying a grudge towards another’s action?) And, you have to accept the mistake you’ve made because you said it voluntarily, and especially if you lose your daughter because you won’t change your ways. Just forgive yourself for the mistake you made (hey, you didn’t harm anyone; you committed no crime if consensual sex, and not paying for sex as a loophole to hiring an escort), but hopefully, be there for her with open arms if and when she has a change of heart and speaks to you--no matter what choice you’ve made—Human beings are never completely predictable. One other thing, however relationship plays out and your daughter maintains communication in the aftermath; trust becomes a factor going forward, and if breached a second time, will it be worth the risk of a good father and daughter relationship coming to an end? It may just come down to how beliefs to morals are interpreted and then play out, but if there is no trust, and lies are found out, then it may be time to walk away from a relationship, which may be your daughter walking away. (Is it worth it?) I wouldn’t beat myself up over the response to your daughter, but there are new things to think about going forward.