Some related threads: http://www.airforums.com/forum...eap-23965.html
It would be hard to imagine anything working short of a pickup truck or something with a standard (not optional) 2" receiver. Newer pickups have the 7-pin plugin but I couldn't imagine it would be easy to equip a rental with a brake controller. And the TV's and an Airstream's pin pattern are different and you'd have to custom wire a replacment plug to the Airstream umbilical.
Years ago I rented a U-Haul and towed a simple trailer with the 4-pin plug -- don't think they'd come with anything like the 2 5/16" ball Airstreams had after the late '60s.
Driving by the large local RV dealer I always see a lot of for sale pickups & SUVs ('gently aged' tow vehicles). Don't know if they'd talk to you about your plan.