Years ago when I had horses and only a car, a couple of times I had to rent a Uhaul truck to pull a horse trailer with. Also happened due to moving when my husband got out of the military and there just were too many things to be pulled and driven.
You can rent either the smallest or the midsize moving truck and it will have a 2" hitch ball on it. They are set up for pulling because people often rent them and a trailer to pull their car behind it. They do allow you to pull a trailer with them, and they do sell a lighting converter so you can hook to the standard lighting of the trailer.
With that said, I suggest the driver be someone who is accustomed to driving large vehicles AND to driving totally with their mirrors. The big downside to doing it this way is that when your vehicle is wider than your trailer it is very hard to judge the distance of the trailer when passing. My husband is a truck driver and has no problem driving and backing a semi trailer, but he finds it MUCH more frustrating to drive a vehicle that is wider than the trailer being towed due to lack of visibility. But yes, you can definitely do it that way.