If you are really targeting a TT it would be worth a call to Johnny Walker RV to ask for a referral. Try Christine DuChesne (702) 249-0507 as she was helping me look at TT's last week.
I'm only aware of JucyRentals.com and the Class C outfits like RoadBearRV.com that do rentals beyond what someone already mentioned about Airstream2Go.com
From what I've read, the Airstream2Go folks would insist on you renting the whole rig as they are very biased towards a turnkey solution (going to the level of supplying a loaner iPad with How-To videos).
There's an outfit (ziontrailerrentals.com) up here by Zion National Park that will put a rental TT onto a space for you, I don't believe they'll let you tow theirs.
2013 Heartland North Trail 22 FBS Caliber Edition
2013 Ford Expedition EL with Tow Package