I had a similar problem with a Ford Explorer. Had the neutral tow kit installed for towing four wheels down behind the camper. It worked as advertised for a number of trips then would not light the indicator lamp when I hooked up for the return trip home.
It was a Sunday and I did not want to chance ruining the automatic transmission so I towed some two hundred miles with the engine in the Explorer running so as to keep the transmission fluid pumping. Never did find out what went wrong as I reverted back to towing the 1997 Saturn SL1 that we still had. Still have the Saturn and still towing it.