We investigate the electronic response of the spin-orbit-coupled Sr₃(Ir_1-xRu_x)₂O₇ system over a wide doping range by using infrared spectroscopy. Our data show that the electronic ground state evolves from the effective total angular momentum J_eff = 1/2 Mott state to a correlated metallic state with Ru doping. We observe that the Ru-doping-driven insulator-metal transition occurs only above a critical Ru concentration of x_c ≈ 0.35. The low-energy optical response of the compound located at the insulator-metal transition boundary shows a peculiar behavior that might be associated with the spin-lattice coupling. Our data also suggest that the effect of the spin-orbit coupling in the electronic response may persist even in the metallic compounds with high Ru concentration. We will discuss the changes in the electronic structure associated with the insulator-metal transition.