The manifestly gauge-invariant and non-perturbatively complete lattice formulation of the weak interactions and the Brout-Englert-Higgs effect is connected to the usual perturbative description in phenomenology via the Fröhlich-Morchio-Strocchi mechanism. However, slight differences between the two have been observed, which can potentially be accounted for by augmenting perturbation theory. We report on our ongoing lattice investigations of these additional effects using a setup with two generations of leptons coupled vectorially to the gauge-Higgs system. We explore the spectrum, inner structure in terms of weak (quasi-)PDFs, and spectral functions of the system to eventually compare cross sections to experimental results.