I can’t imagine a responsible, reputable rental agency renting you a trailer to pull with an Elantra. I can’t imagine a responsible, reputable hitch shop willing to install a hitch capable of pulling that trailer on an Elantra. I wouldn’t feel safe towing much more than a lawnmower with your car.
Best thing to do is rent a U-Haul truck, pack it full, and tow the Elantra behind it.