This is a follow up to a previous Question (Clarification of definition of class $DisNP$)
The class $DisNP$ has a fundamental problem called the (P-separable) Separator Problem. Essentially the question is as follows:
Given any DisNP-pair $(A, B)$. The separator is a Set $S$ such that $A \subseteq S$ and $B \subseteq Complement(S)$. Now, does $(A, B)$ have a separator belonging to $P$? If yes, then $S$ is P-separable else P-inseparable.
The Doubt: I assume the above is an existence problem with an answer 'yes' or 'no'. Now, Given we have an oracle that answers 'yes', can we in polynomial time with a polynomial amount of queries get the actual separator $S$ (similar to creation of the solution of an $NPC$ problem using an $NP$ oracle, that just answers 'yes' or 'no'?