building a polynomial algorithm that solves SAT when given a polynomial TM that solves SAT on two formulas
Here is something your oracle can do: if you are given two formulas $\alpha,\beta$, and are guaranteed that at least one of them is satisfiable, then you can find one of them which is satisfiable by calling $M(\alpha,\beta)$: if $M(\alpha,\beta) = 1$ then $\alpha$ is satisfiable, and if $M(\alpha,\beta) = 0$ then $\beta$ is satisfiable.

Given a satisfiable formula $\varphi$ over $x_1,\ldots,x_n$, you can use this to find a satisfying assignment, variable by variable: at each step, you consider assigning the variable the two possible truth values, and use the method described above to find an assignment under which the formula is still satisfiable. Finally, you can check whether the supposed satisfying assignment indeed satisfies $\varphi$. If so, $\varphi$ is satisfiable. Otherwise, it isn't.