A natural problem in $\textrm{S}_2^\textrm{P}$?
How about a problem complete for promise-$\mathsf{S_2^p}$?

Lance Fortnow, Russell Impagliazzo, Valentine Kabanets, and Chris Umans. On the complexity of succinct zero-sum games. Computational Complexity 17:353-376, 2008. 

From the abstract:


  We prove that approximating the value of a succinct zero-sum game to within an additive factor is complete for the class promise-$\mathsf{S_2^p}$, the "promise" version of $\mathsf{S_2^p}$. To the best of our knowledge, it is the first natural problem shown complete for this class.


(Historical note: it is not too surprising that not many natural problems are known to be in $\mathsf{S_2^p}$ but not known to be in its subclasses $\mathsf{MA}$ or $\mathsf{P^{NP}}$. If you check the original papers of Russell--Sundaram and Canetti (independently), it seems as though the definition of $\mathsf{S_2^p}$ was made more or less specifically to capture their improved arguments placing $\mathsf{BPP}$ in $\mathsf{PH}$, rather than to capture some set of of natural problems.)