Not to contradict Seadaddler but I'm pretty certain that all the AB units are DC powered. Having said that, have you double checked whether 1) your batteries are truly getting charged through shorepower, and 2) whether those batteries are actually holding the charge. What could be possible is that, while plugged into shorepower the pushing the amps through the charger and batteries so your refrigeration would work fine. But when you unplug, that "source" isn't there anymore and there's not enough 'juice' in the housebank to power the compressor.
The first thing I'd do would be to put a meter on all the connections (and check them while your at it!) and make sure the proper 120V AC power is coming out of your charger and into your batteries. If all is good there, do a load test on the house bank and make sure they can charge fully, and then hold that charge.