A voltage reading of 5.5 on a 6 volt battery or 11Volts on a pair in series (or a 12Vbattery) is not low, it is effectively DEAD. It sure sounds like they are not being charged properly. Charging by running the engine for a short period will not help as it takes many hours (12-24 or more) to fully charge a battery. A strong external charger is sometimes needed to at least bring a battery up before these built-in chargers can work effectively on a badly discharged battery.
The easiest way to check the battery charging output is to put the voltmeter across the battery terminals and plug the coach in to 120V. The output across the terminals should be at least 13.2 to 13.4 Volts if the unit is functioning, a similar voltage should be seen while running the engine and charging is taking place. Be sure any battery cutoff switches are in the "ON" position (as opposed to a "Storage" position; although those may not be the exact terms used in that particular coach switch labelling!!). It would also be wise to check any output protection fuses or breakers on the unit itself. Many units have fuses that will blow should the batteries get hooked up accidentally in reverse or overload the charger.
A surface charge is the visible charge that can be measured shortly after disconnecting a charger. It usually disappears within an hour or so, but, if present, it does seem to indicate that some charging was taking place. What you are seeing is, effectively, is a dead battery.
As far as thoroughly checking the battery while connected, I would say NO but it depends on what you expect to be done. They may be able to check the specific gravity and voltage (depending on accessibility) just as you can, but, to do a load test they have to be removed, fully charged and then load tested using specialized equipment for that purpose. An RV repair place is NOT the place to do that, your local battery seller, an automotive parts seller etc can often do it at little or no cost. Once that is done, depending on the results, look again at the converter/inverter.