The pleats sweep better if you sit on the right side. So you can either have someone else drive or move to the UK.

Otherwise, I sweep the best I can as I get seated and then arch my back to raise off the seat and rearange as needed.

My HOE 13oz wool kilt seems to be pretty resistant to wrinkles and they fall out quickly. I drove for three hours in my tank and upon arriving home it looked better than my PV kilt which had been lying across the top of the luggage. I don't know if the pleat depth is a factor but my PV casual wrinkles much more than my tank. After hanging for the night both kilts were wrinkle free the next morning.