We investigate the 5d transition metal oxide BaOsO$_3$ within a combination of density functional theory (DFT) and dynamical mean-field theory (DMFT), using a matrix-product-state impurity solver. BaOsO$_3$ has 4 electrons in the t$_{2g}$ shell akin to ruthenates but stronger spin-orbit coupling (SOC) and is thus expected to reveal an interplay of Hunds metal behavior with SOC. We explore the paramagnetic phase diagram as a function of SOC and Hubbard interaction strengths, identifying metallic, band (van-Vleck) insulating and Mott insulating regions. At the physical values of the two couplings we find that BaOsO$_3$ is still situated inside the metallic region and has a moderate quasiparticle renormalization $m^*/m approx 2$; consistent with specific heat measurements. SOC plays an important role in suppressing electronic correlations (found in the vanishing SOC case) through the splitting of a van-Hove singularity (vHs) close to the Fermi energy, but is insufficient to push the material into an insulating van-Vleck regime. In spite of the strong effect of SOC, BaOsO$_3$ can be best pictured as a moderately correlated Hunds metal.
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