A search for pair production of bottom squarks in events with hadronically decaying $\ensuremath{\tau}$-leptons, $b$-tagged jets, and large missing transverse momentum is presented. The analyzed dataset is based on proton-proton collisions at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ delivered by the Large Hadron Collider and recorded by the ATLAS detector from 2015 to 2018, and corresponds to an integrated luminosity of $139\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. 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 ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{2}^{0}$ and a bottom quark, with ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{2}^{0}$ decaying into a Higgs boson and the lightest neutralino ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{0}$. The search focuses on final states where at least one Higgs boson decays into a pair of hadronically decaying $\ensuremath{\tau}$-leptons. This allows the acceptance and thus the sensitivity to be significantly improved relative to the previous results at low masses of the ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{2}^{0}$, where bottom-squark masses up to 850 GeV are excluded at the 95% confidence level, assuming a mass difference of 130 GeV between ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{2}^{0}$ and ${\stackrel{\texttildelow{}}{\ensuremath{\chi}}}_{1}^{0}$. Model-independent upper limits are also set on the cross section of processes beyond the Standard Model.