Gauge invariance requires physical states to be composite, even in the weak sector of the Standard Model (SM). The Fröhlich-Morchio-Strocchi (FMS) mechanism resolves this subtlety and predicts additional Higgs contributions in SM processes. While this has been supported by theoretical investigations in the bosonic sector, its impact on fermionic observables remains largely unexplored. We use non-perturbative lattice techniques within a gauge-invariant framework to study a proxy theory of the weak sector with dynamical fermions. We determine the physical spectrum of the theory and interpret the results in the context of the FMS mechanism. Additionally, we identify suitable simulation points for a scattering analysis as a first step toward cross-sections relevant to (future lepton) colliders.