Here are two non-examples, one erring in each direction:
Too simple: "The continuum is an $\aleph$-fixed point," that is $\mathfrak{c}=\aleph_\mathfrak{c}$. This is equivalent of course to $\mathfrak{c}\ge\aleph_\mathfrak{c}$, which means it can be expressed as "There is an $\mathbb{R}$-indexed family of sets of reals of pairwise distinct cardinalities," which is $\Sigma^2_2$. Contra my original guess, however, this does have a $\Pi^2_2$ equivalent observed by Farmer S in the comments below (and I'll add his argument here later when I have more time).
Too complicated (so far!): "The continuum is a limit cardinal." This can be expressed in a $\Pi^2_3$ way (which I originally miscounted - thanks to Andreas Blass for bringing this to my attention) as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_3$ equivalent even granting large cardinals. (In particular, note that "The continuum is $\ge$ some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)
Of course, the first example doesn't work, and the second example almost certainly doesn't work. That said, I think this is a good indication that general continuum combinatorics is a good place to look for high-complexity third-order sentences.