I think that we can take it for granted that Dumbledore is not a rational Bayesian. (Nobody is perfectly, but Dumbledore is not even close.) Arguments about a perfect Bayesian tell us what information is present, not what any particular person will actually believe.
But if time turners work based on what information enters the actual beliefs of actual people, rather than on what information is theoretically present, then that makes a big difference. If scenarios A & B are indistinguishable, that wouldn't matter; what would matter is which scenario Dumbledore believes is happening.
If time turners work at the level of the information available through fundamental physics, then it will be impossible to have more than six hours in any overlapping chain of time turners, even ones on opposite sides of the world whose users never communicate in a human way. But probably, they work on something much more like what people actually believe. So your question is actually a very pertinent one.
Elsewhere I note that it is not likely that time-turners would use information-theoretic meanings of 'information' to prevent paradoxes.
I think it's most likely that time-turners simply cannot create causal chains where the effect precedes the cause by more than six hours. (Is it exactly one-hour increments? Which 'hour' exactly, is it- 1/24th of a mean solar day, or is it 33093474372000 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom at rest and 0K?)
Some people have been asking the question about the 6-hour limit on time turners in Harry Potter and the Methods of Rationality. Apparently, if a character, let's say it's Amelia Bones, goes back in time N hours and relays a piece of information about the future to another character, say Dumbledore, then Dumbledore cannot go back in time more than (6-N) hours.
The 6 hour limit is a useful rule to keep the HPMOR universe from becoming over-complicated with time travel, however several people have brought up the following objection. The claim is that Amelia Bones, by traveling back in time and saying that she has information she hasn't yet revealed, has already revealed information about the future in the form of Metadata: That the future still exists N hours in the future and that Amelia Bones was in it, etc. I will show, however, that this is not the case.
Imagine that Amelia Bones has traveled back in time from 1 hour in the future, but she is confused about the time after having apparated across time zones and mistakenly tells Dumbledore that she has information for him from 4 hours in the future. Call this Scenario A. Now imagine that the same scenario happens, but that Amelia is not mistaken about the time. Call that Scenario B.
The thing to note here, is that, from the informational point of view of Dumbledore, provided he doesn't have some additional side-channel information, Scenario A and Scenario B are indistinguishable. (In the same sense that being in an accelerating room is indistinguishable from being in a gravity field in General Relativity.) This is what I mean by "time turner meta-informational relativity." Provided that the act of arriving at some time and place with a time turner doesn't itself leak information about how far in the future you arrived from, meta-information about the future is not the same as information. The time-space coordinate meta-information conveyed when Amelia Bones tells Dumbledore, "I used a Time Turner and I have information about the future," is smeared out over the possible 6 hours. This tells us that meta-information cannot be the same as particular information about the future.
Additional consequence: From this, we can hypothesize that *any* time-indeterminate information conveyed to the past will be "smeared out" over the possible range of times, and that further backwards-time travel is limited by the closest possible value. So, if Amelia came from 3 hours in the future and related a piece of information that leaves it ambiguous if she came from 1 to 5 hours in the future, Dumbledore should still be able to travel 5 hours into the past -- provided he is not also in possession of information that lets him narrow Amelia's possible departure time.
Additional additional consequence: given the above is true, time turners can be used to empirically expose one's possession of such side information. I can imagine this being used for some clever deductive feat.
EDIT: To address some confusion about indistinguishability: 1) This is in the context of a specific point in space-time. 2) A given piece of information can only distinguish Scenario A and Scenario B if it's plausibly consistent with Scenario A but not Scenario B, or vice versa. So the paths of single neutrinos or configurations of air molecules aren't going to be able to do this. However, if there was a leak in a canister of a gas (let's say helium) at the time of Scenario A (4 hours in the future) but not at Scenario B's time, then there would be additional data available in the form of an implausibly large number of helium atoms in Amelia Bone's clothes.