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The doped perovskite BaBiO$_3$ exhibits a maximum superconducting transition temperature ($T_c$) of 34 K and was the first high-$T_c$ oxide to be discovered, yet pivotal questions regarding the nature of both the metallic and superconducting states remain unresolved. Although it is generally thought that superconductivity in the bismuthates is of the conventional s-wave type, the pairing mechanism is still debated, with strong electron-phonon coupling and bismuth valence or bond disproportionation possibly playing a role. Here we use diffuse x-ray scattering and Monte Carlo modeling to study the local structure of Ba$_{1-x}$K$_x$BiO$_3$ across its insulator-metal boundary. We find no evidence for either long- or short-range disproportionation, which resolves a major conundrum, as disproportionation and the related polaronic effects are likely not relevant for the metallic and superconducting states. Instead, we uncover nanoscale structural correlations that break inversion symmetry, which has far-reaching implications for the electronic physics, including the pairing mechanism. This unexpected finding furthermore establishes that the bismuthates belong to the broader classes of materials with hidden spin-orbit coupling and a tendency towards inversion-breaking displacements.