I have a 2015 Subaru WRX that has the same issues with the push button start.
it turns all dash lights and etc on when it is getting ready to tow. This was to go in passive mode after 1 hour and save the battery. It would function correctly parked but if it sensed the wheels turning it would stay on and deplete the battery in two hours. My solution was to install a high end battery and install a charging module .
This solved the discharge problems but it now racks up the miles on the odometer as it is towed.
No solution was provided by Subaru even though it was sold as a towable
