Granular suspensions are intrinsically nonequilibrium systems in which dissipative grain-grain collisions coexist with solvent-induced forcing. We study the particle-scale structure of a granular suspension modeled by smooth inelastic hard spheres immersed in a thermal bath and compare Langevin dynamics simulation results for the radial distribution function g(r) and the static structure factor S(q) with predictions of an equilibrium-inspired rational function approximation (RFA). The equilibrium hard-sphere RFA is supplied with nonequilibrium input for the contact value and a reduced isothermal-compressibility-like quantity, yielding analytical expressions for g(r) in Laplace space and for S(q). We find that the RFA gives a very good description of the short- and intermediate-range structure of the suspension over a broad range of densities, drag coefficients, and inelasticities. It reproduces g(r) substantially better than the Percus-Yevick approximation in inelastic states, especially near contact, and gives a good account of S(q) except at the smallest wave numbers. There, simulations show a drag-dependent enhancement over the RFA prediction, indicating additional long-wavelength nonequilibrium correlations beyond the present equilibrium-like description. These results show that an equilibrium-based hard-sphere approach provides an accurate description of the particle-scale structure of the present Langevin model with inelastic hard spheres (except in the smallest-q region), and suggest that similar equilibrium-inspired approaches may also be useful for related nonequilibrium hard-sphere suspension models, including multicomponent systems.