Pearl takes causality to be primitive, not something to be defined in terms of probabilities. See, for example, "Bayesianism and causality, or, why I am only a half-Bayesian". A basic principle of his methods is that without causal assumptions, no causal conclusions can be obtained: "one cannot substantiate causal claims from associations alone, even at the population level—behind every causal conclusion there must lie some causal assumption that is not testable in observational studies". Such assumptions can be domain-specific knowledge, or they can be general assumptions, such as that the true causal relationships form a DAG possessing the Markov and faithfulness properties.

It should also be noted that Pearl is not God, only a Turing Prize winner. :) It appears that there are disputes within the statistical profession over his methods. I'm not informed on this, but see here for a discussion I came across while trying to track down the quote above.

I don't understand where the tension is supposed to come in. The idea that causation is in the mind, not in the world is part of the Humean tradition and has been a respected (although minority) position in philosophy for centuries. If anything, it seems to mesh particularly well with empiricist leaning philosophies (especially those with an anti-metaphysical stance).

Do you know Jon Williamson's work? It seems to give an answer to your question (but I've not read it yet). Here's the first paragraph of Section 9.1 “Mental yet Objective” of his book “Bayesian Nets and Causality”:

Epistemic causality embodies the following position. The causal relation is mental rather than physical: a causal structure is part of an agent’s representation of the world, just as a belief function is, and causal claims do not directly supervene on mind-independent features of the world. But causality is objective rather than subjective: some causal structures are more warranted than others on the basis of the agent’s background knowledge, so if two people disagree about what causes what, one may be right and the other wrong. Thus epistemic causality sits between a wholly subjective mental account and a physical account of causality, just as objective Bayesianism sits between strict subjectivism and physical probability.

Here's a link to his papers on causality. At least the fifth, “Causality”, contains an introduction to epistemic causality.

1. Probability is "in the mind," i.e., probability claims are true only in relation to some prior distribution and set of information to be conditionalized on;

That statement is too imprecise to capture Jaynes's view of probability. He demonstrates (YMMV) that there is a unique way to assign probability to represent your degree of belief in propositions in a way that is consistent with certain desired properties of degrees of belief. That doesn't make the probability assignment "true", it just makes it consistent with your knowledge and the desired properties. IN particular, it won't make the probability distribution you assign match some ill defined long term frequency of some event occurring.

Hm, good point.

I think the oddness is because good Bayesians usually don't treat the real world as probabilistic, but a pearl-ian model of the world has inherent probabilities. Ideal reasoners can't be automatically right about causal relations, and that's because probability is in two different "places" in the two models. The ideal reasoner is automatically right about its probabilities, but it isn't automatically right about these inherent probabilities in the territory. There's no problem with the probability not corresponding exactly to the world, because that's usually how it is - probabilities are merely the best you can do.

So huh.