Your last post pretty well explained how your inverter is connected. It's on a manual transfer switch with your line from the outside generator connector. From that switch, the current flows to the existing automatic transfer switch, which selects between the rear shore power connector, and the generator/inverter switch. So when you are using your inverter, the power is fed to the main box, and 110 volt power is available to everything in the trailer. Even the air conditioner will get 110 volt power, although there will be no where near enough current to run the air conditioner.
Obviously you will not turn the air conditioner on with the inverter, so it is not really a problem for the air conditioner to have the inverter wired this way.
You will want to be careful not to switch your water heater on electric mode. That will either cause the inverter to shut down, or drain your batteries in no time flat.
In the standard configuration, the converter is powered directly from the breaker without a switch, and will always run when it sees 110 volt power coming to it. As someone else said above, this is not good. The dealer is right, you must always throw that breaker when you run the inverter. It should not be too hard to wire in a simple on/off switch for the converter, and put it in a more convenient place. Then you would not need to touch the breaker.