We describe the effects of finite pressure in a large vacuum facility on a magnetically shielded Hall thruster using internally and externally mounted cathodes. A numerical model of the pumping system was first used to guide the placement of an ionization gauge and an auxiliary propellant injector used for increasing the backpressure. The laboratory model H6MS Hall thruster was operated at 300 and 800 V discharge voltage, 2-20 A discharge current, and 0.6-9.0 kW discharge power at pressures calibrated for xenon from 2.4 to 100 uTorr-Xe. The hollow cathode was located in three positions: internally mounted on thruster centerline and two external mounts outside of the magnetic circuit. Performance variations exceed that expected from ingestion of background gas in all configurations but were still consistent with prior studies of unshielded Hall thrusters. Magnetic field strength, cathode location, and magnitude of discharge oscillations can greatly affect the apparent level of ingestion, which suggests that mechanisms other than ingestion of background gas affect the pressure sensitivity of the discharge. In particular, cathode location is shown to have a profound influence on the measured performance and stability. When the internally mounted cathode is used, the plasma response is shown to be much less sensitive to pressure changes than when an externally mounted cathode is employed. Our results suggest that high-power Hall thrusters using internally mounted cathodes minimize the effects of finite pressure on performance and stability, providing a path for reliably correlating ground tests with flight performance.