But it still remains that in a many circumstances (such as single draws in this setup), there exists information that a Bayesian will find useless and an ambiguity-averter will find valuable. If agents have the opportunity to sell this information, the Bayesian will get a free bonus.

How does this work, then? Can you justify that the bonus is free without circularity?

From a more financial persepective, the ambiguity-averter gives up the opportunity to be a market-maker: a Bayesian can quote a price and be willing to either buy and sell at that price (plus a small fee), wherease the ambiguity-averter's required spread is pushed up by the ambiguity (so all other agents will shop with the Bayesian).

Sure. There may be circularity concerns here as well though. Also, if one expects there to be a market for something, that should be accounted for. In the extreme case, I have no inherent use for cash, my utility consists entirely in the expected market.

Also, the ambiguity-averter has to keep track of more connected trades than a Bayesian does. Yes, for shoes, whether other deals are offered becomes relevant; but trades that are truly independent of each other (in utility terms) can be treated so by a Bayesian but not by an ambiguity-averter.

I also gave the example of risk-aversion though. If trades pay in cash, risk-averse Bayesians can't totally separate them either. But generally I won't dispute that the ideal use of this method is more complex than the ideal Bayesian reasoner.