I had a 2004 3/4 ton Suburban with the 6L and 3.73 axle ratio and had to run 3rd gear towing the trailer in my signature as it would loose speed even on the flats with OD.
As far as I know, all of the GMT800 Denali's had the 6L and 3.73 axle ratio. Look for GT4 on your option codes listed inside the glove box door for the 3.73 axle.
Honestly, from what I know and have read, towing that kind of weight with the 4L60E or 4L65E transmissions I would be doing all of my towing in 3rd.
In regards to gas mileage, my 04 got 15mpg at best on the highway and got 8-9mpg towing. In regards to the fuel pump, I believe it has something to do with the filler neck leading to the gas tank as with our 05 that we had for 5 years I could never run the gas nozzle wide open or it would shut off.