What if the new evidence puts you at even (or close to even) odds? Then you have to live with the cognitive dissonance until further evidence.

There is a difference between managing probability mass and "managing" cognitive dissonance. If one of two events will occur, or only one of two hypotheses can be true, with equal probability according to the information you possess, it is not cognitive dissonance to hold that fact in your head. Cognitive dissonance would be more like formulating two hypotheses in logical contradiction to each other, and insisting they must both be true.

I don't think they are that different. First, the basic insight from Bayes is that belief of all kind can be represented as probabilities. Second, as @dancompton alluded to, most things are not as cut and dry as to say that they are logically in contradiction. There is usually a compatiblist view or a third alternative.

If you are presented with evidence that requires a Bayesian update, and you are smart enough to figure this out, yet you still do not perform the Bayesian update (or you calculate it incorrectly according to your whims as opposed to the evidence), then I would say you are cognitively dissonant in proportion to the change in probability mass you should have recalculated but did not.

Good enough?

So the problem is detecting when that update should be performed, right?

You keep replying to me with one-sentence posts that say little and have no point. If you have something to say or a point you're trying to make, I'd prefer you just do that rather than trying socratic-method everyone to enlightenment.

To (try to) answer your question, to the best of my ability based on limited information what your question even is: no, it is not about detecting when that update should be performed. Remember that we are talking about brain states, i.e. the-world-as-you-think-it-is, as it is represented in your mind, and perhaps if you want to push it, as you are consciously aware of it. If you hold conflicting beliefs but you are truly ignorant of the fact, then you can't really be said to have cognitive dissonance. In Bayes terms, I guess this is analogous to having priors, but not being aware of how to evaluate them based on evidence, and maybe even that you should do so in the first place. Cognitive dissonance, on the other hand, would be having priors, knowing how (and when) to update them, doing the update, and then ignoring the results. Alternatively, it would be performing faulty updates and being at least partially aware of the error in doing so.

It is rare that an arbitrary function negates another for all possible inputs.

If it negates it for the majority of inputs, but you insist it negates it for very few or no inputs, I would call that cognitive dissonance as well. In other words, if it's highly unlikely that two things are both true, but you consider it very likely they are both true, that's probably cognitive dissonance even if the two propositions are not logical contradictions.

Do you disagree that given a bag of all functions which accept N inputs, it is unlikely that 2 chosen at random would negate each other across all N inputs?

Obviously I don't disagree with that. Where are you going with this? People don't formulate their beliefs by picking randomly from a collection of all theoretically-possible beliefs. To expand on your example, if you are selecting from a bag of beliefs according to some criteria (based on some combination of morality, desire, etc) which are themselves in conflict (e.g. "I wish to be feared; I wish to be loved") then I think you are more likely to pick some beliefs which happen to contradict each other (again, if not logically, then for most inputs) than you would just by selecting randomly. Or, probably, if your criteria are more in tune with one another.

How do people formulate their beliefs?

If you are using a Bayesian approach then all new evidence does is adjust the probability of your belief being true. Only those impervious to new information have beliefs with 0 or 1 probabilities.

Cognitive dissonance is not picking X sometimes and !X sometimes, because they're both around 50%.

Cognitive dissonance is saying that X is 90% certain and that !X is also 90% certain.