Let me throw my 6 1/2 cents in.
Your inverters will be different, depending upon model and year produced. For later model year coaches, and Xantrex inverters, the following happens:
When no electric power (incoming 120v) is available (from either shore or generator), and the inverter is on, the inverter will provide 120v power by inverting 12v battery power to 120v. How long that event will last depends entirely upon the amount of 120v appliances demanding power, and especially upon the amount of Amps demanded.
When you connect to shore power or start the generator, the inverter recognizes incoming 120v and stops inverting. The inverter then becomes a charger and charges the inverter's batteries. Few inverters also charge the coach batteries. Most coaches charge the coach batteries via a converter, mounted elsewhere.
The chassis (engine) batteries are charged by the engine's alternator. If your coach sits for a long time, without some sort of disconnect switch thrown, the chassis batteries will drain.
The only downside to having your inverter on constantly, is when you are using shore power, or are running your generator, and either fail for one reason or the other, the inverter switches to provide power to your appliances. You may not realize that your coach is not receiving shore or generator power, and wind up draining your coach's inverter batteries. This is bad, especially when you are going out the door for a day long venture, and the problem is a simple flip of a breaker. You return to dead batteries, and a coach that hasn't had power for a while.
There are a lot of opinions on batteries, ie 6v vs 12v for the inverter, and coach. But when choosing a battery you need to know how much power is in the battery, and how fast (or slow) the battery will give up that power over time. There are formulas to calculate this event.
Just my few cents.
Rick