A search for the exotic decay of the Higgs boson ($H$) into a $bb¯$ resonance plus missing transverse momentum is described. The search is performed with the ATLAS detector at the Large Hadron Collider using 139 $\mathrm{fb}-1$ of $pp$ collisions at $s=13$ TeV. The search targets events from $ZH$ production in an NMSSM scenario where $H\to \chi ~20\chi ~10$, with $\chi ~20\to a\chi ~10$, where $a$ is a light pseudoscalar Higgs boson and $\chi ~1,20$ are the two lightest neutralinos. The decay of the $a$ boson into a pair of $b$-quarks results in a peak in the dijet invariant mass distribution. The final-state signature consists of two leptons, two or more jets, at least one of which is identified as originating from a $b$-quark, and missing transverse momentum. Observations are consistent with Standard Model expectations and upper limits are set on the product of cross section times branching ratio for a three-dimensional scan of the masses of the $\chi ~20$, $\chi ~10$ and $a$ boson.