= Solution
Take
$$
\varphi^*=\varphi^M,
$$
the <formula relativization to a class>: replace each $\exists y\,\psi$ by
$$
\exists y\,(y\in M\land\psi^M)
$$
and each $\forall y\,\psi$ by
$$
\forall y\,(y\in M\mathbin\Rightarrow\psi^M),
$$
leaving atomic formulas unchanged. If $M$ is given by a defining formula, insert that formula wherever $y\in M$ occurs; this is why the map may depend on $M$.
For $z$ and any parameters in $M$, induction on formulas gives
$$
V\models\varphi^M(z)
\quad\Longleftrightarrow\quad
M\models\varphi(z).
$$
Therefore $\varphi^*$-closure puts
$$
\{z\in x:V\models\varphi^M(z)\}
=\{z\in x:M\models\varphi(z)\}
$$
in $M$ for every $x\in M$. This is exactly the <relativized closure criterion for separation>, so $(M,\in)$ satisfies the $\varphi$-instance of separation.
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