Every vehicle has several ratings that relate to towing a trailer. Towing a travel trailer with most half ton and smaller vehicles exceeds some other of the vehicle's ratings. Payload for family RVing is the first thing. If you have 1400 pounds of payload and a family that weights 7000 pounds, then you only have 700 left for trailer TW. Half ton SUVs have payloads that range from 1100-1700 pounds, all with theoretical tow ratings in the 8000+ range. That's great if you are towing a race car on an open trailer by yourself. An RV with family is a lot harder for the tow vehicle. So as you shop, I recommend you check the driver's door for this sticker that is on every 2006+ vehicle. Note what it lists where mine says 2051 for as equipped payload.
That weight trailer is fine for a half ton if the vehicle isn't loaded up with 1000 pounds of people and stuff. Performance wise, you may not be pleased with something with a 4-spd trans unless it was a 2000-2006 8.1L. Any 2009+ half ton 5.3L/6-spd/3.42 would perform very well. Shopping used (or even new) half ton SUVs with the plan for towing a travel trailer can be challenging, especially the 2009+. Many came with the 3.08 axle or 3.42 without the HD trailing package which means under 6000 pound tow rating. Every 3/4 ton includes the extra towing equipment standard. So while rare to find a 3/4 ton, you know it has when you need.
I highly highly highly recommend a 6-spd trans. for GM, that means 2007+ Yukon Denali, 2008+ 3/4 ton, or 2009+ regular half ton. It makes a huge difference IMO in towing performance. If you are open to other brands, 2007+ Sequoia and 2007+ Expedition also come with strong engines and 6-spd transmissions.