Your strategy doesn't work. Deal with it.

I'm afraid you still haven't shown me enough evidence. If you'll bear with me a moment longer, I'll try to explain it better.

But then the mugger can just toss in a Turing machine to his offer, and now your attempt to analyze his offer is equivalent to solving a halting problem

I want you to think of exactly the consequences of that. Now you have two omnipotent god entities, each offering you a more or less random result. All you know about the result is that it's going to be positive or negative. One offers you a random positive result if you kill a baby, one offers you a random negative result if you kill a baby. Do you get, on average and over a very large sample size, more utility from killing the baby or not killing her?