I normally avoid crossing the Big Horns at Sheridan or Buffalo. Especially if I am towing. Your diesel is more than capable, but it is a long steep grade. It is a beautiful drive. I drop south from Custard and take US85 to Lusk, WY, then take US20 up to Thermopolis and then WY120 into Cody. It adds a few miles, but it avoids the Big Horns.
2003 Newmar Scottsdale, SDCA 3257, Workhorse W22, 8.1 Vortec