Here is a typical 30A Electrical Configuration setup.
As you can see the 120VAC and 12VDC Power Distribution Center is the hub for all things Electrical... Also note the trailer SAFETY LIGHTS are not included in this panel. Those 12VDC voltages are supplied by the TOW VEHICLE through the 7-way trailer connector.
The CONVERTER UNIT provide the 12VDC POWER for the the 12VDC side of the Power Distribution Panel. While the CONVERTER is being operated it will also become your BATTERY CHARGER since both units are connected to the same point for providing 12VDC power to the 12VDC Power Distribution Panel.
The Trailer battery also feeds the same 12VDc Side of the Power Distribution.
This is how your trailer can operate using both Shore Power or Generator and Battery.
Shore Power will run ALL of your installed Appliances and systems and without Shore Power the Battery will run only the minimum items and lights to enjoy camping with no electric hookups... You can camp out pretty good with the aid of your propane tanks and Battery bank. This is my favorite way to camp...

If you are going to do alot of camping off grid using your battery it would be a great idea to change out your automotive interiors lights to LED LIGHTS and also increase your battery capacity.
Keep in mind when you are towing your trailer the Working Battery is required by the US DOT when used on PUBLIC ROADWAYS to be able to operate your trailer Electric Bakes in the event the trailer becomes disconnected from your tow vehicle. The on-board trailer battery will send 12VDC to your Electric Brakes stopping your run-away trailer. This is a required DOT SAFETY Feature...
I always refer the the converter as a Converter/Charger when talking about them. Alot of folks still just call the Inverters which is wrong.
Hope this helps some...
Roy Ken