Does a car get heavier as its wheels spin faster, clasically?

A parked car has weight equal to its mass times gravity, in other words, it exerts a force of magnitude mg in the direction normal to the ground, and if you were to weigh the car on a scale, it would show that weight. For a moment, imagine the car at rest in empty space and apply a torque to its wheels. By conservation of momentum, as the wheels begin to spin, the car would begin to spin in the opposite direction. Back on Earth, as the car's wheels begin to spin, the car also tries to spin in the opposite direction, but the ground keeps it from actually spinning, in other words, the car exerts a torque of some magnitude in the direction perpendicular to the ground and the ground exerts an equal and opposite torque that keeps the car from spinning. So the car with spinning wheels is exerting two forces directed perpendicular to the ground, and if you were weigh the car with spinning wheels on a scale, would it show a weight higher than for the parked car?

For this question, please ignore relativity.