You are trying to apply realistic constraints to a hypothetical situation that is not intended to be realistic

Your thought experiment, as you want it to be interpreted, is too unrealistic for it to imply a new and surprising critique of Bayesian rationality in our world. However, the title of your post implies (at least to me) that it does form such a critique.

The gamesmaster has no desire to engage with any of your questions or your attempts to avoid directly naming a number. He simply tells you to just name a number.

If we interpret the thought experiment as happening in a world similar to our own—which I think is more interesting than an incomprehensible world where the 2nd law of thermodynamics does not exist and the Kolmogorov axioms don't hold by definition—I would be surprised that such a gamesmaster would view Arabic numerals as the only or best way to communicate an arbitrarily large number. This seems, to me, like a primitive human thought that's very limited in comparison to the concepts available to a superintelligence which can read a human's source code and take measurements of the neurons and subatomic particles in his brain. As a human playing this game I would, unless told otherwise in no uncertain terms, try to think outside the limited-human box, both because I believe this would allow me to communicate numbers of greater magnitude and because I would expect the gamesmaster's motive to include something more interesting, and humane and sensible, than testing my ability to recite digits for an arbitrary length of time.

There's a fascinating tension in the idea that the gamesmaster is an FAI, because he would bestow upon me arbitrary utility, yet he might be so unhelpful as to have me recite a number for billions of years or more. And what if my utility function includes (timeless?) preferences that interfere with the functioning of the gamesmaster or the game itself?