I'll have to go dig up my manual but I'm pretty sure the 'Tow Mode' instructions say to put it in 3rd when that mode is switched on..
/might have been your trouble all along and why you've bogged so bad on steep inclines, leaving it in OD / 4th and with the torque converter locked up. That and holding it at high RPMs trying to strain upward in speed until the vehicle will kick back up a gear, too.
eta
went and dug up my manual, pgs 4-56,4-57, "You can tow in DRIVE (D). You may want to shift the transmission to THIRD (3) or, if necessary, a lower gear selection if the transmission shifts too often (e.g. under heavy loads and/or hilly conditions" [emphasis added]
and there's a bit about Tow Mode meant to really help in town, heavy loads, and when you are operating in a config where GCWR is over 75% of rating. Which for our 1500 Subs with 5.3L and 3.73 gears is a max trailer wt of 7700# and GCWR of 13,000#. And with 4.10 is 8700# and 14000.
Our factory Class III hitches have a data plate that says GCWR of 12,000#, IIRC.
So I'd say with your stout trailer setup and steep protracted grades you should probably by putting it in 3rd, as things are rigged right now.
I've towed cheap ski boats, pair of jetskis, 14' tandem axle enclosed uhaul crammed full of household goods, equipment trailers, a few thousand pounds of slumpstone block on an open rental trailer over the 4000' passes on the north side of L.A. and into the Sierras with our '99 and '05 Tahoes and the heavier stuff always in Tow Mode and in 3rd on the mountain grades and never had any bogging down troubles. In fact the tandem axle uhaul towed like it wasn't even there. But never at the much higher altitudes that Martin routinely drives.
I'm still thinking a tuneup and some sort of variable tuning ability to temporarily boost engine performance on the hard grades - as well as keeping it in 3rd in Tow Mode - ought to solve his bogging down issues. And thusly the potential harm to the new replacement trans.