I don't think the height of the tow vehicle or hitch has anything to do with trailer sway. Trailer sway is usually caused by improper loading of the trailer. The important thing is tongue weight. More tongue weight and the less chance of trailer sway. But, the tow vehicle must handle the tongue weight of probably 200-300 lbs. Most rental car trailers have hydraulic surge brakes, so no need for the tow vehicle to have electric trailer brakes.
Might consider hauling the truck with the car
