The solenoid to connect the house and chassis batteries is triggered by charging voltage from your alternator (engine) or inverter/charger. 11.6 v isn't enough to overcome internal resistance, so that's not a real charging voltage, perhaps that's the issue. Try unplugging shore cord for an hour or two to allow house batteries to discharge a bit, then hook up shore power again and see how much voltage coming from your converter/charger then to bring house batteries back up.
Bob & Donna
'98 Gulf Stream Sun Voyager DP being pushed by a '00 Beetle TDI