The Curt hitch part # C12068 that you referenced has a trailer towing capacity of 3,500 lbs so it would easily handle a small trailer that has a 700 lb motorcycle loaded in it.
For trailer wiring you'd then want the part # C59236 which is a confirmed fit. For a ball mount part # C45522, 2 inch ball part # C40018, or 1-7/8 inch ball # C40001.
Be sure to check the owners manual of your vehicle for the towing capacity as well. view full answer...
For an aluminum adjustable ball mount that fits all of the criteria you mentioned you would want the part # DTSTBM6800-2S. This has both ball sizes you need, can drop down as low as you need, and has an anti-rattle device built into the ball mount as well. view full answer...