An analog of DP for the second level of the polynomial hierarchy
The definition of the class that you are referring to is interesting.

Originally, the definition of the class $D^P$, as the class of the languages that are the intersection of an $NP$ language and a co-$NP$ language, appeared in [1].
Then, the idea behind $D^P$ was generalized to the Boolean Hierarchy ($BH$) [2], which is a hierarchy between $NP$ and $\Delta^P_2$, and $D^P$ is at the second level of $BH$, i.e, $D^P = BH(2)$.
The definition of $BH$ was subsequently extended to the "Boolean Hierarchy over $\Sigma^P_2$" ($BH_3$) [3], which is a hierarchy similar to $BH$ but located between $\Sigma^P_2$ and $\Delta^P_3$.
In $BH_3$, at the second level, there is the class that you are mentioning.
Hence, in the literature, this class is sometimes denoted by $BH_3(2)$.
However, for the similarity between the definitions of $BH_3(2)$ and $BH(2) = D^P$, pretty soon, in the literature, the notation $D^P_k$ was used to denote the generalization of $D^P$ to upper levels in the polynomial hierarchy, even before a formal definition of it was given.
Therefore, the class you are referring to is usally denoted by $D^P_2$.

Regarding the $D^P_k$-hardness of languages, similarly to other definitions, a language $L$ is $D^P_k$-hard if, for every language $L'\in D^P_k$, there is a polynomial reduction from $L'$ to $L$.
Unfortunately, showing that a language $L$ is $\Sigma^P_k$-hard and $\Pi^P_k$-hard is not enough to show that $L$ is $D^P_k$-hard.
However, if there is a reduction from a $D^P_k$-hard language $L'$ to $L$, then $L$ is $D^P_k$-hard as well.
In this respect, it is helpful what it is said in a comment to the initial question.
In fact, a problem $L$ can be shown to be hard for $D^P_k$ if there is a polynomial reduction to $L$ from the task of recognizing pairs of "yes"-instances of two languages, one $\Sigma^P_k$-hard and the other $\Pi^P_k$-hard.
More formally, given two languages $L_1$ and $L_2$, which are $\Sigma^P_k$-hard and $\Pi^P_k$-hard, respectively, if there is a polynomial reduction $h$ such that $\langle s_1,s_2\rangle\in L_1\times L_2 \Leftrightarrow h(\langle s_1,s_2\rangle)\in L$, then $L$ is $D^P_k$-hard [4, by a generalization of the argument in the proof of Theorem 17.1].

[1] C. H. Papadimitriou and M. Yannakakis. The complexity of facets (and some facets of complexity). Journal of Computer and System Sciences, 28(2):244–259, 1984. DOI: 10.1016/0022-0000(84)90068-0

[2] J.-Y. Cai, T. Gundermann, J. Hartmanis, L. A. Hemachandra, V. Sewelson, K. W. Wagner, and G. Wechsung. The boolean hierarchy I: Structural properties. SIAM Journal on Computing, 17(6):1232–1252, 1988. DOI: 10.1137/0217078.

[3] R. Chang and J. Kadin. The boolean hierarchy and the polynomial hierarchy: A closer connection. SIAM Journal on Computing, 25(2):340–354, 1996. DOI: S0097539790178069.

[4] C. H. Papadimitriou. Computational Complexity. Addison Wesley, Reading, MA, USA, 1994.