A search for pair production of bottom squarks in events with hadronically decaying τ-leptons, b-tagged jets, and large missing transverse momentum is presented. The analyzed dataset is based on proton-proton collisions at s=13 TeV delivered by the Large Hadron Collider and recorded by the ATLAS detector from 2015 to 2018, and corresponds to an integrated luminosity of 139 fb-1. The observed data are compatible with the expected Standard Model background. Results are interpreted in a simplified model where each bottom squark is assumed to decay into the second-lightest neutralino χ˜20 and a bottom quark, with χ˜20 decaying into a Higgs boson and the lightest neutralino χ˜10. The search focuses on final states where at least one Higgs boson decays into a pair of hadronically decaying τ-leptons. This allows the acceptance and thus the sensitivity to be significantly improved relative to the previous results at low masses of the χ˜20, where bottom-squark masses up to 850 GeV are excluded at the 95% confidence level, assuming a mass difference of 130 GeV between χ˜20 and χ˜10. Model-independent upper limits are also set on the cross section of processes beyond the Standard Model.