When towing that much load I would suggest running your lowest HP tune just to help protect the transmission.

I long distance tow a center console boat that is a little lighter and less wind resistant than your TT.

I use tow haul and I will tow in 6th but I keep an eye on a few things.

First is I will not allow the engine to pull hard in 6th gear below 1400 - 1500 RPM.

Which is about 65 mph.

If I have to go slower than that I will manually lock out 6th gear.

Also even running 70 mph if I see a bigger hill coming up I will manually shift down into 5th before I get to the hill so that the shift is already completed and the torque converter is already locked back up before I get to the hill.

The truck has plenty of power to pull virtually any hill in 6th with my boat but I'm trying to help the tranny live.

I'll also manually downshift before getting hard on the throttle to step out and pass someone if I'm going to floor it Ill use 4th gear.