When hooked to shore power, most inverter/chargers ONLY keep the house batteries charged, NOT the chassis batteries. If the supplied batteries are AGM batteries, you need to make sure that the inverter/charger is properly set-up for AGM batteries, since they are different from wet cell batteries. You will also probably need to supply a separate battery charger for the chassis batteries.
When the coach is not being used, there is still a drain on the batteries from the CO detector, propane detector, radio memory, smoke detector, etc. All of these things will gradually deplete your batteries unless they are maintained by a battery charger.