This assumption is wrong. You are assuming that the mugger is also a utilitiarian, so will do cost-benefit analysis, and thus decide not to mug. But that is not necessarily true.

If the mugger mugs anyway, despite mugging being "suboptimal," Bentham ends up in a situation where he has exactly the same choice: either lose $10, or have the mugger cut off their own finger. If Bentham is to follow (act-)utilitarianism precisely, he must pay the mugger $10. (Act-)utilitarianism says that the only thing that matters is the utility of the outcome of your action. It does not matter that Bentham previously committed to not paying the mugger; the fact is, after the mugger "threatens" Bentham, if Bentham does not pay the mugger, total utility is less than if he does pay. So Bentham must break his promise, despite "committing" not to. (Assuming this is some one-off instance and not some kind of iterated game; iteration makes things more complicated.)

(In fact, this specific objection -- that utilitarianism requires people to "give up" their commitments -- is at the foundation of another critique of utilitarianism by Williams: https://123philosophy.files.wordpress.com/2018/12/bernard-wi...)

If everyone were a utilitarian, then there would be far fewer objections to utilitarianism. (E.g. instead asking people in wealthy countries to donate 90% of their income to charity, we could probably get away with ~5-10%.) Bentham's mugging is a specific objection to utilitarianism that shows how utilitarians are vulnerable to manipulation by people who do not subscribe to utilitarianism.

Also, to be precise, Bentham's mugging does not show a contradiction. It's showing an unintuitive consequence of utilitarianism. That's not the same thing as a contradiction. (If you want to see a contradiction, Stocker has a different critique: https://www.jstor.org/stable/2025782.)