Can stratification be used to internalize functions on models of $\sf Z$?

Question

Suppose $M$ is a model of $\sf Z +\neg AC$ that is externally bijective to an element $k \in M$. Obviously if $j$ denotes such an external bijection, then it cannot be used in Separation within $M$, because the range of $j$ from $\mathcal P^M(k)$ would be a subset of $k$ that is bijective to $\mathcal P^M(k)$ by $j^{-1}$, and using Separation one can easily recover the diagonal which would be an element of $M$, and thus $M$ would no longer be bijective to $k$.

Would it be safe to use $j$ in stratified instances of Separation within $M$? Where "stratified" is defined after Quine with the addition that $j(x)$ and $x$ receive the same type during stratification.

The same question but in terms of $\sf ZF$ was posed here and found to be inconsistent. Here, is a possible salvage.

Answer 1

Let T=kxP(k). By stratified Separation with j, there is a set S such that

S consists of the pairs (ab)∈T such that j(a)=b. S is an onto function from a subset of

k to P(k) which is impossible.