If physics beyond the Standard Model (BSM) explains the $4.2\sigma$ difference between the Standard Model and measured muon anomalous magnetic moment $a_{\mu }$, minimal flavor violation predicts a shift in the analog quantity for the $\tau$ lepton $a_{\tau }$ at the ${10}^{-6}$ level, and even larger effects are possible in generic BSM scenarios such as leptoquarks. We show that this produces equivalent BSM deviations in the Pauli form factor $F_2\left(s\right)$ at $s=(10\mathrm{GeV}{)}^2$ and report the first complete two-loop prediction of $\mathrm{Re}{F}_2^{\mathrm{eff}}\left(100{\mathrm{GeV}}^2\right)=-268.77\left(50\right)\times {10}^{-6}$ for resonant $\tau$-pair production in $e^{+}{e}^{-}\to \Upsilon \left(nS\right)\to {\tau }^{+}{\tau }^{-}$, $n=1$, 2, 3. $\mathrm{Re}{F}_2^{\mathrm{eff}}$ can be measured from $e^{-}$-helicity-dependent transverse and longitudinal asymmetries in $\tau$-pair events, which requires a longitudinally polarized $e^{-}$ beam. We discuss how Belle II asymmetry measurements could probe $a_{\tau }^{\mathrm{BSM}}$ at ${10}^{-6}$, assuming such a polarization upgrade of the SuperKEKB $e^{+}{e}^{-}$ collider, and conclude by outlining the next steps to be taken in theory and experiment along this new avenue for exploring realistic BSM effects in $a_{\tau }$.