Plausibly, some true set theoretic axioms can be known a priori with a high degree of confidence. The Axiom of Separation is a good candidate. But other axioms, like the Axiom of Choice or the Continuum Hypothesis, we are much less confident of. Suppose that these axioms are true.
Could smarter beings than we are know them a priori? Maybe, but probably even they would not know them with complete confidence. There will be arguments for the axioms and arguments against the axioms. It seems likely that there will be axioms of set theory that both are true but that no rational being is going to have an a priori degree of confidence greater than, say, 0.99. Moreover, it is likely that there are many such independent axioms, perhaps infinitely many. When you conjoin enough such axioms, the probability of the conjunction will get smaller and smaller for a rational being, until you get to the point where the conjunction will be a priori quite incredible. Thus, there will be conjunctions of a priori knowable axioms that will themselves not be a priori knowable.
Thus the a priori knowable is not closed under conjunction. This does not bother me. As Jon Kvanvig said to me, the a priori is an epistemological category, and so we shouldn't expect closure. But I think this will generate serious problems for anybody like Chalmers who wants to put a heavy philosophical burden on the concept of the a priori.
But maybe someone could have a rational insight into set theory that yields complete certainty as to a controverted axiom, of a sort that remains no matter how many independent axioms are conjoined? For instance, theists are apt to think that God has such an insight. But God is not, I think, an a priori knower of set theory. First, I say that abstract objects are nothing but divine thoughts, and so God knows set theory by introspection, and introspection is more akin to the a posteriori. Second, even apart from such an ontology of sets, it is really hard to see if the rational insight really should count as a priori. I don't know how the rational insight would work, either for God or for a godlike knower, but rational insight into set theory is something like a vision of set theoretic reality. But that's much more like the a posteriori.