Either U-haul didn't ask or you told them you were putting a smaller car on their trailer because I know they wouldn't allow that size/weight vehicle to be towed on their trailer.
Check with some places that sell trailers if they rent car haulers.
I went to buy a new car hauler and after talking to the salesman about how much I needed to use, not often, he suggested renting from them.
I can rent a like new trailer, 18' deck, 2' dove tail, 16" tires with spare and electric trailer brakes for last time I rented, $35.00 day.