I'm still inclined to think the tires are out of round. Balancing will not fix it. I've seen tires flat-spot from sitting in one position too long (very common on classic vehicles), even sitting upright in the tire store racks. In the old days tire shops had shavers that would true tires by taking rubber off the high side, but the only place I know that still shaves tires are some truck/motorhome alignment shops.
