Our 10 Edge tows fine and the battery doesn't run down with the charge wire and Roadmaster tow bar except when wife leaves the switcch in the accessory notch and not all the way back. With the ignition in the accesory spot, notch or whatever you have electricity routed from the car's battery to the accessories and lights and the wire from coach out of the circuit.
We talked with someone in Savannah with an 11 Edge with the push button start and they encountered the same problem. They followed the then current manual, which seems to have left out something and the battery was dead after towing.
This subject has been encountered before here but I can't recollect a solution.