Finding a minimal DFA whose language has a desired intersection with another
$M_C$ must accept every word of $S^+ = B$ and reject every word of $S^- = A \setminus B$.

Let $A$ and $B$ be finite and such that both $S^+$ and $S^-$ are non-empty. Then exact computation of $M_C$ is NP-hard. [1]

[1] E.M. Gold. Complexity of Automaton Identification from Given Data. Information and Control, 37, 302-320 (1978)