I do not think you will be happy long term with the Korean offerings. As much as I had enjoyed my Hyundai Genesis and Sonata, within two years (even with no warranty dealer visits) they just did not ride the same as new. I deduce Towing 4,000 lbs will age them faster. And I agree with you on the CVT transmissions - always searching for something not there. The design was to improve the curve on federal fuel consumption requirements with no regard for driver frustration.
Here is a curve ball that before automatically elminating I would recommend you research and take a test dirve.
Take a look at the Jeep Grand Cherokee Eco Diesel. It offers 6,000 lbs towing with fantastice fuel mileage. I know of two people who have the exact same engine in their RAM 1500 trucks and they love them. Quiet, too. On the road they are happy to hum along. They are totally the opposite of the Cummins diesel that announces its arrival a block away. You get the Grand Cherokee ride (stick with 2WD if you don't need 4WD - suspensions are different with the 2WD and much smoother). Jeeps reliability, thanks to Mercedes input during their marriage, just keeps on improving. Now with Fiat financial backing, Dodge, RAM, Chrysler and Jeep will lead the pack soon.
Second option, as I often talk about, would be to consider a VW Touareg TDi. I have personal experience with them. They have the diesel torque to dry dock a battleship. The only way it comes is with AWD so the fuel consumption will never beat a 2WD. Still, however, expect 30 mpg when not towing. And not far off when towing. If you happen to find yourself offroad, not many SUV's are better it has such a fine drivetrain (check out YouTube and watch some of the videos of its capability). When you sit in a Touareg you will start smiling at the fantastic ergonomic design. Again, this is a very quiet engine. Even at idle, most will not realize it is a diesel. Diesel fuel is more expensive, but every time I did the math on expense vs mileage the diesel won.
Do some research on the new, clean diesels. You will be very surprised how far they have come. Standing behind the vehicle at the exhaust you will never smell anything that resembles the old smelly ones of years ago. They burn clean diesel fuel - so clean that they emit less emissions into the atmosphere than many gas engines of today. Both come with factory setups for towing.
Either one of these would be a win-win for your listed needs.