Yes I understand that 3^^^3 is finite. But it's so unfathomably large, it might as well be infinity to us mere mortals. To say an event has 1/3^^^3 is to say you are certain it will never happen, ever. No matter how much evidence you are provided. Even if the sky opens up and the voice of god bellows to you and says "ya its true". Even if he comes down and explains why it is true to you, and shows you all the evidence you can imagine.
The word "negligible" is obscuring your true meaning. There is a massive - no, unfathomable - difference between 1/3^^^3 and "small" numbers like 1/10^80 (1 divided by the number of atoms in the universe.)
To use this method is to say there are hypotheses with relatively short descriptions which you will refuse to believe. Not just about muggers, but even simple things like theories of physics which might allow large amounts of computing power. Using this method, you might be forced to believe vastly more complicated and arbitrary theories that fit the data worse.
If you apply Solomonoff induction to predict arbitrary integers, you get undefined expectations.
Solomonoff inductions predictions will be perfectly reasonable, and I would trust them far more than any other method you can come up with. What you choose to do with the predictions could generate nonsense results. But that's not a flaw with SI, but with your method.
Summary: the problem with Pascal's Mugging arguments is that, intuitively, some probabilities are just too small to care about. There might be a principled reason for ignoring some probabilities, namely that they violate an implicit assumption behind expected utility theory. This suggests a possible approach for formally defining a "probability small enough to ignore", though there's still a bit of arbitrariness in it.