The problem here is that even if Scenario A and Scenario B are indistinguishable, Amelia's words still constitute Bayesian evidence on which Dumbledore can update his beliefs.

In my formulation, that's "side information." Really, my gedankenexperiment doesn't work unless Amelia Bones happens to be very ditzy concerning time.

I'm inclined to believe that whatever intelligence is behind capital-T Time is enforcing an intuitive definition of information, in the same way that brooms work off of Aristotelian mechanics.

So then, this is a limitation in the "interface" that the Atlantean engine is following. I think my hypothesis is testable.