Ultimately, to the degree that you value their life in comparison to yours.
This is the selfish answer, not the utilitarian answer. (I think the selfish answer is stable.) Note, though, that you have some choice over how much you value their life- someone who strongly values autonomy might decide to get angry at being held hostage to reduce the amount they value the other person, making it easier for them to break up with them.
But if you're less happy in your relationship than your long-term background level, the correct move is to break up now, even though that will push you both even lower in the short term.
Why doesn't the long-term background level depend on the other relationship options you have? As you say later, the correct comparison is not singledom!
It does depend on your other options, but people are very bad at estimating the value of their other options. They idealize potential partners they don't know well, they overestimate others' interest in them ("She looked at me for slightly longer than she looked at him! That must mean something!"), and so on. An "outside view" estimate based on either (a) what it was like to be single or (b) your previous long-term background including past relationships is less vulnerable to bias, and likely to be more accurate.
Breaking up in order to ...
Utilitarianism seems to indicate that the greatest good for the most people generally revolves around their feelings. A person feeling happy and confident is a desired state, a person in pain and misery is undesirable.
But what about taking selfish actions that hurt another person's feelings? If I'm in a relationship and breaking up with her would hurt her feelings, does that mean I have a moral obligation to stay with her? If I have an employee who is well-meaning but isn't working out, am I morally allowed to fire him? Or what about at a club? A guy is talking to a woman, and she's ready to go home with him. I could socially tool him and take her home myself, but doing so would cause him greater unhappiness than I would have felt if I'd left them alone.
In a nutshell, does utilitarianism state that I am morally obliged to curb my selfish desires so that other people can be happy?