With 30 amps you will have plenty of charging for your batteries. The inverter/charger will be charging the coach batteries, but not necessarily the chassis batteries. With shore power connected, the "inverter" part will be inactive, but if the shore power fails, it will quickly draw down the batteries, even without any AC loads connected, so have the "inverter" side turned off.
To ensure that the chassis batteries stay charged up you need to have either an "echo" charger or "Bird" connected and operating, especially if your coach is in covered storage and the solar panel on the roof is inactive. If you have neither, get a separate "smart" charger for the chassis batteries, or at least disconnect the batteries from the ECM or its constant draw will quickly exhaust those batteries.
The fridge (assuming it is the original Norcold or Dometic) runs on AC power or propane, except for the control circuits which use 12 volt DC power from the coach batteries. I wouldn't leave the fridge on for 2 months. At best it will need defrosting and its contents will be old and dried out. At worst, it will be off when you return and all of your food will be spoiled.
So wiith everthing turned off, you would likely get away with no shore power, but hey, you have 30 amps, so all will be well when you return.
