Pb$_2$CoOsO$_6$ is a newly synthesized polar metal in which inversion symmetry is broken by the magnetic frustration in an antiferromagnetic ordering of Co and Os sublattices. The coupled magnetic and structural transition occurs at 45 K at ambient pressure. Here we perform transport measurements and first-principles calculations to study the pressure effects on the magnetic/structural coupled transition of Pb$_2$CoOsO$_6$. Experimentally we monitor the resistivity anomaly at $T_N$ under various pressures up to 11 GPa in a cubic anvil cell apparatus. We find that $T_N$ determined from the resistivity anomaly first increases quickly with pressure in a large slope of $dT_N/dP$ = +6.8(8) K/GPa for $P < 4$ GPa, and then increases with a much reduced slope of 1.8(4) K/GPa above 4 GPa. Our first-principles calculations suggest that the observed discontinuity of $dT_N/dP$ around 4 GPa may be attributed to the vanishing of Os magnetic moment under pressure. Pressure substantially reduces the Os moment and completely suppresses it above a critical value, which relieves the magnetic frustration in the antiferromagnetic ordering of Pb$_2$CoOsO$_6$. The Co and Os polar distortions decrease with the increasing pressure and simultaneously vanish at the critical pressure. Therefore above the critical pressure a new centrosymmetric antiferromagnetic state emerges in Pb$_2$CoOsO$_6$, distinct from the one under ambient pressure, thus showing a discontinuity in $dT_N/dP$.
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