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The trimer lattice BaIrO₃ is a spin-orbit-coupled Mott insulator with a Néel temperature of T_N =183 K. We report structural, magnetic, transport, and thermal properties of single-crystal BaIrO₃ with dilute Bi doping for Ba or Ba_1-xBi_xIrO₃ with x ranging from 0.001 to 0.01. While retaining the native crystal structure, mere 0.3% Bi doping or x = 0.003 precipitates a wide range of drastic reductions in the resistivity by 5 orders of magnitude, and the magnetic moment (0.01 μ_B/Ir) and the Seebeck coefficient (~20 μV/K) by one order of magnitude. Furthermore, the Sommerfeld coefficient (e.g., 7 mJ/(mol*K²)) and the Wilson ratio (e.g., 2.4) consistently demonstrate an emergent metallic state that is both robust and correlated. The high susceptibility of the ground state to dilute Bi doping is largely because of the Ir₃O₁₂ trimers whose subtle alterations dictate the physical properties. The results will be discussed along with comparisons drawn with related iridates.