Let $\kappa$ be measurable cardinal(or other large cardinals) and let $j:V\longrightarrow M$ witnessing this. We know that $j(\kappa)$ has large cardinal properties in $M$, but what about $j(\kappa)$ in $V$.

Let me give a couple tentative nonstandard definitions:

$\bullet$ Call a measurable cardinal $\kappa$ weakly compact preserving, if there is $j$ an elementary embedding witnessing measurability of $\kappa$, such that $j(\kappa)$ is weakly compact in $V$.

$\bullet$ Call a measurable cardinal $\kappa$ reflecting preserving if $V_\kappa\prec V_{j(\kappa)}$, again in $V$.

What I am really interested in is a weakly compact preserving-reflecting preserving measurable cardinal $\kappa$ such that there is a weakly compact cardinal $\lambda>\kappa$ such that $V_\lambda\prec V_{j(\lambda)}$.

Question: Where such a cardinal take place in large cardinals hierarchy?