Can a NEXP machine simulate invalid queries to a promise problem oracle?
Let's define what P$^A$ means in the promise setting: A language L is in P$^A$ if there is a polytime oracle machine $M$ such that for all sets $B$ such that $B$ contains $A_{YES}$ and $B\cap A_{NO}=\emptyset$, $L=L(M^B)$. In other words, the polytime machine can ask queries that are not part of the promise but its output cannot depend on whether those queries are answered yes or no.
For example, consider unique SAT where $A_{YES}$ are the formula with exactly one satisfying assignment and $A_{NO}$ are the formula with no satisfying assignment. Then $P^A$ is contained in $\oplus P$. We also have $NP\subseteq BPP^A$ by Valiant-Vazirani. Both these tools are used in the proof of Toda's theorem.