Complexity class for NP with access to a DP oracle
You have not found explicit mention of it because it is the same class as $\Sigma_2 = \textbf{NP}^\textbf{NP}$. $\Sigma_2 \subseteq \textbf{NP}^\text{DP}$ holds because any $\textbf{NP}$ query can be poly-time (many-one) reduced to a SAT query, which, in turn, can be cast as a DP query (by setting $\varphi_2$ to a trivially unsatisfiable formula). Conversely, $\Sigma_2 \supseteq \textbf{NP}^\text{DP}$ because any DP query can be answered by two (adaptive) SAT queries.