We set our generator to automatic just to be safe and not draw the batteries down too low.

When we go on a trip we turn on the inverter main switch (outside), turn on the inverter  control/display module (inside), turn on our residential fridge and use as normal from that point on.  The inverter and batteries provide enough power for the fridge while we travel, which can be up to 7 hrs at a time.  We’ve never boondocked or dry camped and used the fridge, so I don’t know just how long the batteries would last without shore/generator power....probably depends on usage.  The fridge uses almost no power most of the time, but uses a lot I would guess when the compressor and fan kick on.  The only other draw I can think of are like the lights when you open the door and the LED display.  When we are done with a trip we make sure we turn off the inverter main switch in the outdoor compartment, because even if the fridge is off the inverter control/display module draws enough power itself to eventually drain the batteries.

We have a 2017 Fleetwood 32A Storm, 35 foot.

At home I plug into shore power all the time. I'm a novice as far as an inverter is concerned. I turn off the inverter from inside the RV and notice the battery charging system is still functional. It keeps the charge of the 4 golf cart batteries to about 13.2 volts (2 each 6 volt batteries in series) for a float charge. If the batteries get too low then it automatically begins to charge. However when the big red switch in the inverter  compartment is switched off the batteries don't get a float charge and it shows a voltage of about 12.8. So I leave that switch on. Would that be the thing to do if shore power is always connected at the house?