Solomonoff induction does need to be modified here to support multiple observers, but the modification is minor; we just need to allow a universe to specify a probability distribution over observers (this is basically the SSA; different modifications are needed for different anthropic theories). The results in the probability of each individual bitstring including a factor of 2^-1000000, but the summed probability - the probability of "QM is (approximately) true" - is much greater than the probability of any alternate explanation. Most of that probability mass comes from algorithmically random strings, so that is what you should expect.
You're about to flip a quantum coin a million times (these days you can even do it on the internet). What's your estimate of the K-complexity of the resulting string, conditional on everything else you've observed in your life so far? The Born rule, combined with the usual counting argument, implies you should say "about 1 million". The universal prior implies you should say "substantially less than 1 million". Which will it be?
EDIT: Wei Dai's comment explains why this post is wrong.