If the rig is always hooked up to shore power, does that mean that the generator is not used to actually power anything when it is run? Or do you disconnect shore power while running the generator?
If it is always hooked to shore power, my guess is that the converter/charger is always on the batteries. If so, I'd guess that the problem is that that converter/charger is NOT one of the newer 3 stage units, but one of the cheaper/older two stage ones that have a tendency to kill the battery over time if just left charging all the time. If this is so, the batteries are truly dead and will need replacement, and I would add a battery cut-off (assuming your converter can be run without a battery) and I'd hook the batteries to a BatteryMinder or BatteryTender to keep them charged and desulphated.