As an act utilitarian, the utilitarian was trying to evaluate the consequences of the act, not a rule that could be followed in multiple instances. Therefore, credibly claiming that the act will be a secret removes any consideration of motivating other people or being judged by other people, etc. (Missing from the story was a promise by the mugger not to repeat this with the utilitarian every day).

I don't see why the utilitarian should trust a mugger's promises of secrecy or non-replicability though?

The mugger is a Deontologist in this scenario and therefore does not lie. If the utilitarian couldn't trust the mugger's promises, the whole scenario would fall apart as they couldn't trust the mugger's promise to cut off their finger.

How does the utilitarian know this?

Any morality needs to take into account our uncertainty about claims other people make.

The mugger has a lapel pin denoting himself as a deontological agent. Lapel pins in these fantasies cannot be forged, I guess.

If we're assuming unforgeable moral-method pins I don't think we should expect intuitions generated in this sort of thought experiment to be a good guide to what we should actually think or do.

The mugger is a deontologist, right? We're already assuming that he'll keep his promises.

I mean, he's a mugger. Clearly doesn't take his deontology too seriously.

But notably not threatening or committing violence toward anyone else!

No, the mugger getting the money counts as negative. "Now, as an Act Utilitarian, I would happily part with ten pounds if I were convinced that you would bring more utility to the world with that money than I would. The trouble is I know I would put the money to good use myself – whereas you, I surmise, would not."

No, it doesn't. People having money is Good under utilitarianism because they can utilize it no matter which person it is.

Utilitarianism does not benefit from covert insertions of specific moral carve-outs. Surmisal does not impact outcomes only predictions of outcomes. It is not appropriate to make judgments based on surmisal because utilitarianism can only ever look backward at effects to justify actions post-hoc. This is the primary flaw with utilitarianism as a moral philosophy.

I'm also confused why they drop this point. I don't give in to this kind of threat because I expect overall a policy of giving in leads to worse outcomes.

Act utilitarians specifically don't believe in evaluating the overall consequences of a policy. Rule utilitarians do that. That is, in fact, the major difference between the two.

Good point, I phrased it poorly. Because of the effects of the specific action, I think an act utilitarian should still refuse to be mugged in this case.

The policy I was describing is just a mental shortcut, a part of adapting morality to human beings. See https://en.wikipedia.org/wiki/Two-level_utilitarianism for more in this direction.

Two level utilitarians are primarily rule utilitarians, not act utilitarians.