If the battery charger is adequate, it shouldn't allow the batteries to drain when connected to shorepower. If it is not adequate, you could always replace the charger with a higher-current output unit. That will set you back a few hundred dollars, but much less than a new fridge.
Having said that, if the fridge is otherwise in good condition, simply add a 120VAC to 12VDC @ 10AH power supply,(or whatever the current draw of the fridge is, plus at least 25% for margin). Isolate the power supply output from the batteries using a battery isolator.
I would imaging you could do that for probably $100. +/-, and your fridge will automatically switch between 120VAC and 12VDC. A new fridge will set you back close to a grand, not counting the additional fabrication work involved.