Upward absoluteness of well-foundedness fails for transitive models of Zermelo set theory (i.e. with full Separation but no Replacement). That is, you can find $M \subseteq N$ both transitive models of Zermelo so that there's a linear order $L \in M$ which $M$ thinks is a well-order but $N$ sees is ill-founded.
This follows from Theorem 2.2 of Harvey Friedman's 1973 paper "Countable models of set theories" (https://doi.org/10.1007/BFb0066789). Let me state an instance of the theorem suitable for this question.
Let $\alpha$ be a countable admissible ordinal and let $T$ be a theory in the language of set theory which extends KP so that $T$ has a transitive model with $\alpha$ as an element. Then there is an ill-founded model of $T$ whose well-founded part has height exactly $\alpha$.
(Briefly: You can prove this using the Barwise compactness theorem. Doing it for $T$ being ZFC: Let $U$ be a countable model of ZFC with $\alpha \in U$ admissible. Then $A = \mathrm L_\alpha$ is an admissible set. You can cook up a theory $E$ in the infinitary language $\mathcal L_A$ whose models must be end-extensions of $A$ whose well-founded parts have height $\alpha$. (Here, $Y \supseteq X$ is an end-extension if for any $a \in X$ if $Y \models b \in a$ then $b \in X$.) Then $U$ witnesses that $A$-finite fragments of $E$ + ZFC are consistent, and so by Barwise compactness $E$ + ZFC has a model.)
Let $\bar M$ be a model of ZFC obtained from this theorem for countable admissible $\alpha > \omega$. ZFC proves that $\mathrm V_{\omega + \omega}$ is a model of Zermelo set theory, and set $M = {\mathrm V_{\omega+\omega}}^{\bar M}$. In Zermelo, well-foundedness is a $\Pi_1$-property. These are downward absolute among end-extensions. (That's really the same fact that $\Pi_1$-properties are downward absolute among transitive models.) So everything in $M$ which $\bar M$ thinks is well-founded is also thought to be well-founded by $M$. In particular, everything $\bar M$ thinks is a countable ordinal is isomorphic to a linear order on $\omega$ in $M$, which $M$ thinks is a well-order. Fix such a linear order $L$, isomorphic to an "ordinal" $\lambda$ in the ill-founded part of $\bar M$.
Now observe that $M$ is well-founded—because $M$ has height $\omega+\omega < \alpha$ and $\bar M$ is well-founded below $\alpha$. So $M$ is a transitive model of Zermelo which thinks $L$ is a well-order. But $N = \mathrm V_{\omega+\omega}$ sees that actually $L$ is ill-founded. This is because $N$ contains every real and so has a witness to the ill-foundedness of $L$. Thus the desired pair $M \subseteq N$.
