From what I can tell about your complaint, it sounds like your engine alternator is not charging. The best way to test the alternator voltage output is to charge-up all batteries, unplug shore power, start the engine and run it while checking the voltage at the house batteries and the chassis batteries. The voltage should be 13.8 to 14.2 volts on both sets of batteries. If the batteries are not fully charged the voltage will be a little less depending on how low their state of charge is. If the voltage is the same on both sets of batteries with the engine running or not, check the voltage on the alternator terminal (usually the center one) of the isolator. If that voltage is low (12v or less) or 0, you have a bad connection or a bad alternator. If one battery bank is low and one is high. you have a bad connection between the isolator and the batteries or a bad isolator. If both battery readings are low but the alternator terminal is high, you have a bad isolator or both connections from the isolator to the batteries are bad.
The inverter does not charge the chassis batteries. The chassis batteries can only be charged by the inverter when all batteries are connected together (hold down the boost switch or turn the key to the left on your coach). I hope this helps.
Gerald