A search for pair production of bottom squarks in events with hadronically decaying $\tau$-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$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. 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 $\chi ~20$ and a bottom quark, with $\chi ~20$ decaying into a Higgs boson and the lightest neutralino $\chi ~10$. The search focuses on final states where at least one Higgs boson decays into a pair of hadronically decaying $\tau$-leptons. This allows the acceptance and thus the sensitivity to be significantly improved relative to the previous results at low masses of the $\chi ~20$, where bottom-squark masses up to 850 GeV are excluded at the 95% confidence level, assuming a mass difference of 130 GeV between $\chi ~20$ and $\chi ~10$. Model-independent upper limits are also set on the cross section of processes beyond the Standard Model.