First-- Do you have a Magnum ME 2000 or the Magnum ME 2012 inverter charger? I am unable to find a ME 2000 inverter on an internet search.
Second-- How long were you plugged into the 15amp shore power? Even though you have a 100amp 3 stage charger, it still takes 6 to 12 hours to get your batteries fully charged.
Third-- You say you turned your gas elect fridge to elect after you disconnected from shore power. Are you really sure the fridge was actually working on elect. Gas/elect fridge's should not be wired through the inverter. All these fridges I know of only have 3 power settings, Off, Gas only, and Auto. If the fridge is on the auto setting, it will default to gas if there is no 120V AC present, which there should not be if not plugged into shore power and the generator is not running.
As long as you were plugged into shore power long enough, 6-12 hours, to charge your batteries, they should not have dropped to 12.2V in 90 minutes.
If your fridge indeed is wired though the inverter, do this test again and after 90 minutes turn off the fridge and check your voltage. Having a load on your batteries will cause them to show a lower voltage than the batteries would at rest. However the load from the fridge should not drop the voltage from the fully charged voltage of 12.6V to 12.2V in 90 minutes. Not with over 400AH of battery.
Also as you noted, when connected to shore power you are seeing 13.6-13.8 volts. When you disconnect from shore power you should still see the 13.6V for some few to several minutes. It should then decrease to about 12.6V which is the voltage of a fully charged battery bank.
If you continue seeing 12.2V you should take a multi meter, and check the voltages of your individual batteries. I'm assuming you have four 6V deep cycle golf cart batteries. You could have one bad battery.
Have you checked the water level in the batteries? If the water level in any of the battery cells has dropped far enough to expose the plates to the air, that is likely the cause of your problems.