(Here, the log(n) is needed to specify how long the sequence of random bits is).
You don't always need log(n) bits to specify n. The K-complexity of n is enough. For example, if n=3^^^^3, then you can specify n using much fewer bits than log(n). I think this kills your debunking :-)
O(BB^-1) (or whatever it is) is still greater than O(1) though, and (as best I can reconstruct it) your argument relies on there being a constant penalty.
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.