For example, I can simply modify AIXI with a rule that says "if you've seen a sequence of increasingly large numbers that can't be explained by any short computable rule, put some weight into it being BB(1)...BB(2^n)... (and also modify it to reasoning symbolically about expected utilities instead of comparing numbers) and that will surely be an improvement over all possible uncomputable universes. (ETA: Strike that "surely". I have to think this over more carefully.)

This is what I dislike about your logic. You create a situation where (you think) AIXI fails, but you fail to take into account the likelihood of being in the situation versus being in a similar situation. I can easily see a human seeing a long series of ones, with some zeros at the beginning, saying "ah-ha, this must be the result of a sequence of busy beavers", when all he's actually seeing is 3^^^3 minus his telephone number or something. AIXI can lose in really improbable universes, because it's designed to work in the space of universes, not some particular one. By modifying the rules, you can make it better in specific universes, but only by reducing its performance in similar seeming universes.