This is the semantic problem that you dismissed. When I talk about the refrigerator, it's clear that I mean to draw an imaginary boundary around the refrigerator only and pretend for a second that that is all there is anywhere. Then the entropy is decreasing. If I talk about the process by which I acquired that knowledge, then I have to expand my imaginary boundary to include the source of the photons that bounced off the refrigerator, for instance, and the waste heat my brain produced to acquire this knowledge. That process, the acquiring of the knowledge, was entropy increasing even if what it revealed to me was a less entropic distribution over states of the refrigerator.
The refrigerator is the two gas system with a pump attached. Learning anything about either system is an entropy increasing proposition (if the boundary is drawn around me plus the system). As it happens, if you want to draw the boundary to exclude me, then the two-gas-system-without-pump also happens to be entropy increasing, while drawing a boundary around the refrigerator is entropy decreasing.
This seriously is just Maxwell's demon.
As it happens, if you want to draw the boundary to exclude me, then the two-gas-system-without-pump also happens to be entropy increasing...
This is what I'm disputing you can get if you treat entropy as subjective uncertainty, while also assuming that the only way to update subjective uncertainty is Bayesian conditionalization. Perhaps you can explain how the two-gas system turns out to be entropy increasing on that viewpoint if you draw the boundary to exclude the observer. How does the entropy of the probability distribution describing the system increase?
Link to the Question
I haven't gotten an answer on this yet and I set up a bounty; I figured I'd link it here too in case any stats/physics people care to take a crack at it.