You should have greater than 12.2V (or 12.5V) when on shore power with the batteries charging from the converter. The battery voltage and the converter voltage should be the same, excepting any (hopefully fairly negligible) voltage drop due to resistance in the wiring. 1.5V is not reasonable for wiring losses; it indicates an open circuit or at the least a high-resistance connection.
Do you have 12V power when disconnected from shore power? I'm going to guess you do not. It would be the exact same problem, only with the current not flowing in the opposite direction.
I assume you've verified that the monitor reading of 12.2V is a reasonably accurate representation of the actual voltage at the battery (by checking with another meter).