When he says that it's a problem to have two different probabilities from the same situation, he doesn't realize that it's a problem for Bayesian only if the two calculations starts from the same prior information.

It's kind of impossible to prove theorems about when Bayesians should agree without knowing that.

And I don't see the problem with the sentence you quoted, unless you claim that each way encodes different priors (and even so, that would be an answer to the problem, not a reason to say the problem doesn't deserve a response).