Weak null maximum and integration of families of multiplication operators

Let $X$ be a reflexive Hardy space or weighted Bergman space on the unit disk in the complex plane. For a bounded linear operator $S$ on $X$, let $\textrm{wem}(S):= \sup_{(f_n)} \limsup_n \|Sf_n\|$, that is, the supremum of cluster points of $n\mapsto \|S f_n\|$, where $(f_n)$ is any unit norm weakly null sequence. This quantity coincides with the essential norm on the reflexive weighted Bergman spaces. For a suitable family $\{ g_t : t\in]0,1[ \}$ of bounded analytic functions on the unit disk, we characterize when one can exchange $\textrm{wem}(\cdot)$ and integration over $t$ of the multiplication operators $M_{g_t}$, that is, when $\textrm{wem}( \int M_{g_t}\, dt ) = \int \textrm{wem}( M_{g_t} ) \, dt$; when the functions $g_t,t\in]0,1[$ can be continuously extended to the unit circle, we obtain a neat function-theoretic characterization.