Paging Dick_B... He has experience with the 8.1L and now the 6.0L/6-spd. I believe his prior TV was a Suburban which did not use the Allison trans, but experience should be similar. I looked for an 8.1L Burb for about a year before buying a new 6.0L/6-spd. That is one strong engine.
Going to 6-spd with the 6.0L will feel quite a bit better. Take a look at this chart comparing the gearing and torque multiplication for the SUV 4-spd trans to the 6-spd. The Allison does a little better with a 3.1:1 first gear ratio. As far as I know, all pickups with 8.1L used the Allison trans and the Suburban/Yukon XL/Avalanche 8.1L had the in house 4-spd as Burbman mentioned.
So the newer 6.0L still likes to rev. It would still be singing or screaming at 5000rpm up some hills. The 6-spd will help get and keep the engine in it's happy range. You would probably end up spending most of your time towing in 4th gear as I don't know that it could hold 5th on flat highway. You need the 4.10 axle as the tow ratings for 6.0L/6spd/3.73 is in the 9500-ish pond range. 4.10 is about 12k until 2011 I believe which is near 14k pounds.
Honestly, at your 12k estimated weight, I would suggest 8.1L or the diesel you said you don't want.
A & A parents of DD 2005, DS1 2007, DS2 2009
2011 Suburban 2500 6.0L 3.73 pulling 2011 Heartland North Trail 28BRS
2017 Subaru Outback 3.6R
2x 2023 Chevrolet Bolt EUV (Gray and Black Twins)