Does PV=nRT mean that isobars are also isotherms?
No. After a bit of mathematics, and the inclusion of water vapor, you can also get $P=\rho_d R_dT_v$, where $\rho_d$ is the dry air density and $R_d$ is the specific gas constant for dry air, and $T_v$ is the virtual temperature. So isobars are more than just isotherms- they are also dependent on density and water vapor. Check out Wikipedia's derivation for the proof that $PV=nRT$ is the same as $P=\rho RT$, and then use the fact that $\rho$ has both dry air and water vapor, to get $P=\rho_d R_dT_v$.