I'll try to engage with the text.

Take this part of the exchange:

>BENTHAM. Fair enough. But, even so, I worry that giving you the money would set a bad precedent, encouraging copycats to run similar schemes.
MUGGER. Don't. This transaction will be our little secret. You have my word.

Say we accept the Mugger's claim. Wouldn't the Mugger repeat the threat of self-harm immediately after? The previous 10£ is already lost. What matters to Bentham is the future, so his moral calculus would be the same. That is, on the second iteration of the threat, Bentham must hand over another 10£. And so on...

So we run into an infinite regress. Say Bentham considers this option already at the first threat. He then does a calculus that assigns a non-zero probability to that the threat will continue until Bentham is required to give the Mugger more than what a finger is worth (assume a finger has finite utility). Bentham can then rationally reject the threat on basis that accepting the threat, we end up on a slippery slope that after some finite number of threat iterations of finite time leads to negative global utility.

The Mugger could then commit to that if given 10£, that would be the end of said threats of self-harm, to Bentham or anyone else. Now wouldn't we say that Bentham is right to fork up the cash? I'd say so. 10£ seems like a small price to pay to make the Mugger give up all future threats of self-harm (we assume here the Mugger is truthful). So maybe the Mugger should up the price -- why ask for a measly 10£ if it only can be done once? And now we are in familiar, but arguably unavoidable, "icky" territory of assining cash value to the physical well-being of individual humans. What price should Bentham (or the community) be willing to spend in order to prevent a certain self-harm of a certain individual among us? And we are back in territory where utilitarinism works pretty well.

So.... how well did I do? I think these one-shot moral dilemmas are tricky in that the reason they often conflict with our intuitions is because we think about the next step, the step after that etc. If the Bentham stand-in truly could know, or at least expect (given assigned probabilities) that there is one finite cost to accept for the removal of a negative utility, then our moral intuitions would line up with utilitarianism: there is a cash value we should be ready to depart from when that act removes a quantifiable moral ill.