What would be $p(A\cap B \cap C)$ using the multiplicative Bayes theorem?
Is the multiplicative Bayes' Theorem any different from the plain old Bayes' Theorem? The Bayes' Theorem I know is not about $p(A\cap B)$; instead, it relates $p(A\mid B)$ and $p(B\mid A)$. So I wouldn't expect Bayes' Theorem to say anything about $p(A\cap B\cap C)$. You might find this Wikipedia page helpful. It does include some formulas involving $p(A\cap B\cap C)$.