A search for the pair production of higgsinos $\tilde{\chi}$ in gauge-mediated supersymmetry models, where $\tilde{\chi}$ decays to a light gravitino $\tilde{G}$ either via a Higgs or $Z$ boson, is presented. The search targets final states in which a Higgs boson decays to a photon pair, while the other Higgs or $Z$ boson decays to a $b\bar{b}$ pair, with missing energy associated with the two gravitinos, and it is performed with the ATLAS detector using 139 fb$^{-1}$ of proton--proton collisions at a centre-of-mass energy of $\sqrt{s}$ = 13 TeV. Search regions dependent on the amount of missing energy are defined by the requirements that the di-photon mass should be consistent with the mass of the Higgs boson, and the $b\bar{b}$ mass with the mass of the Higgs or $Z$ boson. The main backgrounds are estimated with data-driven methods using the sidebands of the di-photon mass distribution. No excesses beyond Standard Model expectations are observed and higgsinos with masses up to 320 GeV are excluded, assuming a branching fraction of 100\% to $h\tilde{G}$, partly covering a previous gap in sensitivity in this regime.