I would suggest a Jeep Grand Cherokee with the 3.0 Ecodiesel or the Hemi. These SUV's have 7,400 lbs tow rating and are very capable. The V6 Pentastar has a tow rating of 6,200 lbs. These Jeeps get 26-30 mpg highway (not towing)
Talked to a man this weekend who was towing his 5,000 ish lb Airstream with a Pentastar Grand Cherokee. The rig looked like a good match and he said it was a "good riding" combo. He was using a standard weight distribution hitch without any sway control.
If you want to spend a few more bucks the Mercedes GL450 has a good tow rating too.
There are several SUV's with about 5,000 lb tow ratings. Here is a short list
Nissan Pathfinder
Kia Sorento V6 AWD
Chevy Traverse
Ford Explorer
Honda Pilot
Hyundai Santa Fe
If you need more tow rating the best choices are going to be the Chevy Suburban (I got 22 MPG out of a 2015 rental on the highway) or the Ford Expedition/Lincoln Navigator Ecoboost twins.
Thanks and good luck!
Jeremiah