Topological polaritons provide a route to engineer optical Chern bands, but realizing large gaps or efficient electrical tunability remains challenging because time-reversal-symmetry breaking in semiconductor polariton systems is typically limited by weak Zeeman splittings. Here, we demonstrate a topological Bose-Fermi mixture — Fermi-polaron polaritons ---where large gaps and electrically invertible topology emerge from the interplay of Fermi-polaron many-body interactions, photonic symmetry, and strong light–matter coupling. Using a gate-controlled monolayer transition-metal dichalcogenide integrated with a symmetry-engineered photonic crystal, we create Fermi-polaron polaritons with topological band structures. The direct coupling between band topology and gate-tunable many-body interactions provides a mechanism for amplified time-reversal-symmetry breaking beyond bare Zeeman splitting and enables electrical inversion of topological gaps, Berry curvature, and Chern numbers. These results establish an electrically reconfigurable platform for interaction-driven topological states and programmable topological photonics.