I'm looking for the following:

(1) an example of a $\Pi^1_1$ class-theoretic sentence that has no known equivalence to a $\Sigma^1_1$ sentence, even if large cardinal hypotheses or reflection principles are assumed.

(2) an example of a $\Pi^1_2$ class-theoretic sentence that has no known equivalence to a $\Sigma^1_2$ sentence, even if large cardinal hypotheses or reflection principles are assumed.

Admttedly, "large cardinal hypothesis" and "reflection principle" are not well-defined, so these questions are rather vague.

The best answer I currently have for (1) is Vopenka's principle ("every class of directed graphs has two distinct elements that are connected by a homomorphism"). But this is often considered a large cardinal hypothesis, so it's not a great answer.

As for (2), I have no answer at all.