I wouldn't expect the trans TSB to correct anything heat related. If you were saying you were having issues shifting into 3rd then I could maybe see the TSB correcting it, but it sounds like gravity's talons were dug deep into your trailer!

I went through a similar situation in my '00 when I moved out to PA from Phx. I was dragging an overloaded U-haul trailer with my truck (37" tires and 3.73 gears) up along the Mogollon Rim. My truck was wrapped out in 1st gear, just had the pedal mashed for about the same length of time you did. "Back then" though, you didn't have any dash messaging, you waited for the idiot gauge to instantly peg or the trans to blow up.

Ah the good ol' days!