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 fb⁻¹ of pp collisions at s = 13 TeV. The search targets events from ZH production in an NMSSM scenario where H → χ~20χ~10, with χ~20→aχ~10, where a is a light pseudoscalar Higgs boson and χ~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 χ~20, χ~10 and a boson. [Figure not available: see fulltext.]