Where did 3^^^3 pop out of? Outside of mathematics, "always" never means "always", and in the present context, Omega does not have to be perfect.
Allowing for a margin of error, the simulation would indeed make do with lower fidelity. Yet the smaller the margin of error that is tolerable, the more the predictive model would have to resemble / be isomorphic to the functionality of all components involved in the outcome ((aside from some locally inversed dynamics as the one you pointed out).
Given an example such as "chess novice versus grandmaster", a very rough model does indeed suffice until you get into extremely small tolerable epsilons (such as "no wrong prediction in 3^^^3 runs&qu...
Just developing my second idea at the end of my last post. It seems to me that in the Newcomb problem and in the counterfactual mugging, the completely trustworthy Omega lies to a greater or lesser extent.
This is immediately obvious in scenarios where Omega simulates you in order to predict your reaction. In the Newcomb problem, the simulated you is told "I have already made my decision...", which is not true at that point, and in the counterfactual mugging, whenever the coin comes up heads, the simulated you is told "the coin came up tails". And the arguments only go through because these lies are accepted by the simulated you as being true.
If Omega doesn't simulate you, but uses other methods to gauge your reactions, he isn't lying to you per se. But he is estimating your reaction in the hypothetical situation where you were fed untrue information that you believed to be true. And that you believed to be true, specifically because the source is Omega, and Omega is trustworthy.
Doesn't really change much to the arguments here, but it's a thought worth bearing in mind.