I think that the converter might draw to much current on start up. I would think about making an adapter to connect your generator 12v line to the tow vehicle pig tail (12v leads only) to charge the battery. This would require a trailer receptical to match your trailer plug. You would not need to remove the battery from it's mount just unplug from the tow vehicle and plug into the adapter.
Steve
64 Tradewind
KC7TJQ