There are a few medium sized SUV's with tow ratings above 7,000 lbs, including the VW Touareg, Jeep Grand Cherokee, and Range Rover Sport. I wouldn't hesitate to tow up to 5,000 lbs with any of those.
If that's not enough trailer for you then a Ford Expedition with towing package will do 9,200. Chevy Tahoes, Dodge Durangos, and Nissan Armadas are a little less but still at least 8,500. I'd probably do up to 6,000 lbs with any of those.
Any more than that and I'd go for a truck with a crew cab. Our F350 gasser had a huge back seat that was bigger than most SUV back seats. Plenty big enough for 2 dogs back there, and the vinyl + rubber floor was super easy to clean. Just blow it out with a leaf blower after a trip.