I tow with a Jeep Grand Cherokee and this is WAY TOO MUCH. Not only is it too heavy, it is TOO LONG.
The JGC is a great tow vehicle (apologies and respect to Les Adams), but only in the mid- to light-weight ranges. In the proper weight range, you can turn the TT brakes off and the rig will not even stutter when braking. But the JGC is not a 3/4-ton dually, nor does it want to be. Find a different rig to transport your TT with, and enjoy it, but not with the JGC.
I'm not the weight police nor will I suggest doing it, but if you're set on towing the Airstream with your Cherokee I would strongly suggest talking to Andy at Can Am RV. He is considered to be the guru of custom hitches and an expert on the Airstream set ups. People come to him from all over the world. Check out the website for some of his set ups and contact info.
'07 Toyota Tundra Crewmax Limited
'06 Forest River Flagstaff 26BH(sold)
'13 Keystone Bullet Premier 31BHPR
Hitched by Hensley, Viewed by McKesh
Airstreams, tow better than most BUT I think you are asking for trouble. Rent a TV and have someone drive your jeep, turn in the rental in florida and you only pay one way. JMHO
Do you have a link to a rental source nationwide that allows towing items other than what they rent and isn't a box type truck.
I can never remember seeing any rental vehicle towing a POV TT. Maybe they exist, but I have my doubts on finding one.
Arond here ENTERPRISE RENT A CAR has full size trucks with reciever hitches on them, A few years ago they even had a few diesels. I use to work at a body shop that did this regions repairs. Try them they are nationwide.