Lots of good advice posted above!
You may consider looking at a Ford Expedition. Different models in the line have different towing capacities. Chevy has a similar SUV.
Actual weights are what matters. Load up your Jeep and TT rig and go to a CAT scale. For $15 you get axle weights. Drop the trailer in the parking lot and reweigh the Jeep ($15 more). Subtract the Jeep weight without the TT from the Jeep weight with the TT. The difference is the tongue weight. Now you have actual weights.
Jeep front axle
Jeep rear axle
Total Jeep weight
TT axle total
Total combined weight
Next copy down the specifications from the stickers in the Jeep driver's door frame. There may be two stickers. One for the Jeep and one for the jeep tires. The rear hitch will have its own rating. Maximum cargo weight may be listed, if not then (MGVW - Dry or Unloaded weight) is Cargo weight.
Any specifications listed there must not be exceeded when the TT is attached.
MGVW Maximum Gross Vehicle Weight
Maximum front axle weight
Maximum rear axle weight
Jeep and TT Combined gross vehicle weight
Maximum towing capacity
The trailer also has a weight specification sticker on the front driver's side. MGVW must not be exceeded. Specified tongue weight is usually not correct. Ignore it.
Actual cargo weight is the specified dry weight subtracted from the actual total weight of the loaded Jeep with the TT attached.
Tongue weight including the weight distribution hitch must not exceed specified hitch maximum. Tongue weight needs to be at least 10% of trailer weight for safe towing. 15% will be more stable.
As posted above, distance between front and rear wheels of the tow vehicle matter. When traveling faster than 45 MPH, longer wheel base matters. It reduces sway caused by light tongue weight, passing trucks, and cross winds. The faster you go, the more pronounced the effect.
Use the above bench marks for evaluating possible new tow vehicles. To get more stability, you need more margin between actual and maximum than the Jeep gives you. If any Jeep specs have been exceeded, reduce speed and leave a lot more stopping distance in front of you. Get a higher capacity tow vehicle.
Factory installed tow options often include more enhancements than you may expect. So look for them. Otherwise there is may be much more to learn.
Look at the door sticker for any new or used vehicle you may buy. Do this before committing to buy. Run the numbers. Individual vehicles in the same model line differ substantially. Half ton trucks are not all the same. They vary a lot.
I have a 21 foot Dutchmen Kodiak Cub ultra-lite. I tow it with a 2015 Nissan Pathfinder AWD SUV with factory towing package. The actual weight of the Cub is about 4300 #. The Nissan is spec'ed for towing 6000 #. Driver, 1 passenger, 1 dog, luggage is within limits. Tongue weight between 500 and 750 #.
Towing at 65 MPH is stable including climbing Midwest hills and emergency stopping. This is the maximum I would tow with the my Pathfinder.
I have looked at a Ford Expedition when I was considering towing an Airstream 27 foot TT. There is also a Chevy with similar towing capacity. Individual vehicles capacity vary a lot. Always check the door stickers before committing to buy.
I wish you good luck and happy trails ahead!