A search for pair-production of vector-like leptons is presented, considering their decays into a third-generation Standard Model (SM) quark and a vector leptoquark ($U_1$) as predicted by an ultraviolet-complete extension of the SM, referred to as the '4321' model. Given the assumed decay of $U_1$ into third-generation SM fermions, the final state can contain multiple $τ$-leptons and $b$-quarks. This search is based on a dataset of $pp$ collisions at $\sqrt{s}=13$ TeV recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of up to 140 fb$^{-1}$. No significant excess above the SM background prediction is observed, and 95% confidence level limits on the cross-section times branching ratio are derived as a function of the vector-like lepton mass. A lower observed (expected) limit of 910 GeV (970 GeV) is set on the vector-like lepton mass. Additionally, the results are interpreted for a supersymmetric model with an $R$-parity violating coupling to the third-generation quarks and leptons. Lower observed (expected) limits are obtained on the higgsino mass at 880 GeV (940 GeV) and on the wino mass at 1170 GeV (1170 GeV).