The following is well-known. If $\kappa$ is measurable, $\theta > \kappa$, and $M \prec V_\theta$ has size $<\kappa$, then there is $N\prec V_\theta$ such that $N \supseteq M$, $M \cap \kappa \not= N \cap \kappa$, but $N \cap \sup(M \cap \kappa) = M \cap \kappa$. Repeating this a number of times allows us to also find such $N$ with the ordertype of $N \cap \kappa$ being any ordinal $\leq \kappa$.

The construction of $N$ can be done by adjoining a single ordinal that is in all measure-one sets in $M$ for some normal ultrafilter, and closing under Skolem functions. If $M$ is $\mu$-closed, then so is $N$. However, repeating this $\omega$-times in an elemenatary chain can kill countable closure.

My question: Is there is some large cardinal $\kappa$ with the following stronger property where we demand closure?

Whenever $\mu<\kappa$ is regular, $\theta > \kappa$, and $M \prec V_\theta$ is $\mu$-closed and of size $<\kappa$, then for every $\alpha <\kappa$, there is $N \prec V_\theta$ such that $N$ is $\mu$-closed, $N \supseteq M$, $M \cap \kappa \not= N \cap \kappa$, $N \cap \sup(M \cap \kappa) = M \cap \kappa$, and the ordertype of $N \cap \kappa$ is $\geq \alpha$.