A search is performed for dark matter particles produced in association with a resonant pair of Higgs bosons using 140 fb$^{−1}$ of proton-proton collisions at a centre-of-mass energy of 13 TeV recorded by the ATLAS detector at the Large Hadron Collider. This signature is expected in some extensions of the Standard Model predicting the production of dark matter particles, and is interpreted in terms of a dark Higgs model containing a Z′ mediator in which the dark Higgs boson s decays into a pair of Higgs bosons. The dark Higgs boson is reconstructed through final states with at least three b-tagged jets, produced by the pair of Higgs boson decays, in events with significant missing transverse momentum consistent with the presence of dark matter. The observed data are found to be in good agreement with Standard Model predictions, constraining scenarios with dark Higgs boson masses within the range of 250 to 400 GeV and Z′ mediators up to 2.3 TeV.