Let $\kappa$ be a measurable cardinal (or other large cardinal) and let $j:V\longrightarrow M$ be a witness. 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:

Call a measurable cardinal $\kappa$

*weakly compact preserving*, if there is an elementary embedding $j$ witnessing measurability of $\kappa$, such that $j(\kappa)$ is weakly compact in $V$.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: What is the place of such a cardinal in the large cardinals hierarchy?