If you know that $\kappa$ has high Mitchell rank, then it follows that there must be many measurables below with high Mitchell rank, since this is a $\Sigma_2$-property, observable in any model with the same $V_{\kappa+2}$. So it reflects below.

For example, if $j:V\to M$ has critical point $\kappa$ and $M$ has all the measures on $\kappa$, then $M$ will notice that $\kappa$ has high Mitchell rank, and so it is true in $M$ that there are such measurable cardinals below $j(\kappa)$, so there must be many such measurable cardinals below $\kappa$ in $V$.

The answer is yes. If you know that $\kappa$ has high Mitchell rank, then it follows that there must be many measurables below with high Mitchell rank, since this is a $\Sigma_2$-property, observable in any model with the same $V_{\kappa+2}$. So it reflects below.

Specifically, if $j:V\to M$ has critical point $\kappa$ and $M$ has all the measures on $\kappa$, then $M$ will observe that $\kappa$ has the Mitchell rank that it does. So it will be true in $M$ that there is a measurable cardinal as desired below $j(\kappa)$, and consequently by elementarity there will be many such measurable cardinals below $\kappa$ in $V$.