You need to use the TX and RX wires for port 3. Confusingly enough I believe in the RDI/AGRemote setup, it might refer to this as Port C. Serial Port 2 (or Port B) always defaults to TSIP when the receiver is reboot, if I'm not mistaken. But if you set Port 3(C) to OUTPUT NMEA and INPUT RTCM, that should work with CORS. If CORS requires NMEA for resolving the nearest VRS, then make it GGA only, and does not need to be more than 1 hz.
What you said is factual in the vast majority of applications. If receiver is used for auto steering, which is 99.999% of the applications for rtk unlocked 372, port b will reset to tsip on boot. If used standalone port b will not “reset”. As you stated, port c (found on the same physical connection) is the “safe” route for external corrections.
Trimble systems always have some sort of “gotcha” and little consistency in firmware versions. Example, port D of fmx used to be able to output nmea . Version 6 firmware broke that feature for those using autopilot but you can still get nmea while having an ezsteer. Huh? Oh, btw, it will let you select it in the menu. It just doesn’t work. No documentation.