The first way, AS connected directly to generator, will charge your batteries if your Store/Use switch is in the Use position. You will be charging your batteries with your converter (a single stage battery charger), so don't leave it charging indefinitely or you'll ruin your batteries.
The second way, external battery charger connected to generator, will charge your batteries. If your external battery charger is a three-stage charger, then you're good to go. The position of your Use/Store switch will affect the background load on the batteries, so it's best that you keep the Use/Store in the Store position to minimize the parasitic load.
The third way, battery charger connected to generator and AS connected to generator, will work. If the Use/Store is in the Store position, you'll be charging the batteries with your external battery charger and the converter will supply 12 V to the rest of the trailer. Don't switch the Use/Store to the Use position since both your external charger and converter will be competing to charge your batteries.
2015 30' Classic, "Chez Nu"
2014 RAM 2500 w/Cummins Diesel
ProPride Hitch, 400 W Solar