I remember seeing in the owners manual that Hyundai does not recommend towing with the Genesis, HOWEVER, the car has plenty of power for towing, and seeing as Drawtite make a hitch for it, they must have found strong places to attach the hitch. In your case, a Motorcycle Trailer would be no problem. I used to tow an enclosed single bike trailer with many of my cars, including my Continental (Ford says no more than 1000 lbs,) and my set up (Enclosed Chariot model 10-6) with the bike inside, weighed about 2000 lbs.
It could be that Hyundai wants to protect the Transmission. The V8 (ZF) Transmission can probably take the extra load better than the V6 (Aisin) Transmission, but don't know for sure.
I don't know if the Genesis, either transmission, has a transmission cooler or not. (Not the type in the radiator, but the type that sits near the front of the radiator, in the air flow). If you are going to tow any distance, in very hot temperatures, I would add an additional Transmission Cooler, in front of the A/C Condenser. My Continental has an external transmission cooler, so I never added an additional cooler. In previous cars, I always added the
aftermarket cooler, connected so that the fluid went to the additional cooler AFTER it left the stock cooler. My automatic cars were always connected this way, be it a 4cylinder Toyota, an 8cylinder
Buick or a 6cylinder Chrysler. Never had transmission problems with any of them.