Joel,
Assume for a moment that there are two legs coming off the salesman switch, one to the lights and one to another load that has some current draw on it going somewhere, say the circuit board in the fridge and the solenoid valve safety device for the propane system and that is the CMP sense line. In theory you are right that there can be just one voltage on the output side of the relay and if you have no load on the light circuit and measured it, it would read the same as the relay terminal. If there was load on the other circuit that the CMP measurement point would see then it would have a lower voltage at its measurement point. It is hard to diagnose compare the two voltage wise without knowing the loads. The issue that adds some "fog" to this is that the CMP readings change as you reset the relay and seem to recover and I see no way that the wiring path to the CMP would coincidently heal itself for a resistive wiring path issue going to the CMP because of resetting the relay. There is no doubt that the hand held voltmeter reading coinciding with the inverter panel is incorrect. Just not clear how resetting the relay can cause the CMP to recover. It is probably not worth a lot more effort on this.
The voltage supply line that is the sense line for the CMP is probably what supplies the unit 12V in the cabinet. You could find that and measure that point with your voltmeter and see what the voltage difference is between the two and have your answer as to the accuracy of the CMP and what voltage the line is seeing with relay resets.
Later Ed
