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Ba_1-xK_xBiO₃ exhibits a plethora of remarkable phenomena, such as charge-density wave (CDW) order,  a metal-insulator transition, and high-temperature superconductivity. While the average structural symmetry of this perovskite has been documented in detail, crucial questions pertaining to local structural symmetry and short-range order remain unanswered. Here we present a semi-phenomenological Monte Carlo study of several models of the local structure of Ba_1-xK_xBiO₃ near the metal-insulator transition. The models are based on various hypotheses about the underlying physics and chemistry and informed by diffuse x-ray scattering data. In this way, we provide a framework to qualitatively test various hypotheses and for interpretation of these data. This allows us to answer some of the long-standing questions regarding remnant short-range CDW order, the interplay of frustrated tilts and polar distortions, and the possibility of local symmetry breaking in Ba_1-xK_xBiO₃ .