I would suggest that you look at vehicles with a 6 speed transmission. The 10th generation you mention are 4 speed transmission. 4 speed is fine on the flats, but in the mountains the extra gears are important.
As you are probably aware, gasoline engines make the horsepower at pretty high RPMs. When traveling in the mountains/hills, where you are likely towing at slower speeds, it's hard to keep the RPMs high enough in the lower gears to make the necessary horsepower to move the load.
I know you mention not wanting a pick-up. We had that same thought early, mainly because of 3 kids and a dog, so we started towing our jayFlight 26BH (essentially the same TT as you) with a '04 Tahoe. We were right at the cargo capacity limit as we only had ~100 lbs to spare while packing very light and not taking extras (bikes, kayaks, firewood etc). We shopped in vein for a 2500 suburban. Couldn't find a decent one at a reasonable price, ended up reluctantly going with a 2500 crew cab pickup. Three years later it was the best decision we could have made. We now take all the toys we want, ended up getting a larger TT to support our family more comfortably and we still have cargo capacity to spare.
Unfortunately the automobile manufacturers no longer produce many (if any) larger SUVs capable of being a sufficient tow vehicle for mid-large TT when you also have a growing kids.