A 2005 should be 4AT so towing in 3rd means it just alternates between 3rd and 2nd. Those 4L60’s had big jumps between gears. Not sure if in 2005 if the owners manual "recommended" towing in Drive or Overdrive, used to be, on pretty much all 4AT's, one was to lock out OD--and if you really do have 3.42's, I'd use Drive too.I have an 05 Suburban 5.3, Awd, w/ air ride and towing package. I think it has 3.42 gears. I pull a 5k lb camper trailer around the south east and smoky mountains. My weight distribution setup is spot on. It is my opinion that this set up is just adequate. The suburban has good towing manners and fits the whole family. I do feel that the 5.3 is underpowered. I only travel at ~65 MPH and struggle on inclines to maintain speed. I don't use cruise control as it will downshift and rev to the 4.5k+ on any incline. In comparison, my buddy pulls an identical camper with his 2017 Ram 1500 with the hemi. He doesn't have these issues. I also manage 9-10 MPG when towing. 14-17 MPG with no load.
A 2500 suburban would be ideal!
[Side note: look for the build sheet. Used to be on the glovebox, not sure on this vintage. I think GU6 is 3.42:1. Find the RPO code and that will say what you have, unless if you bought used and/or swapped gears.]
On the flip side, letting the speed droop on hills means more mpg. More hp means more gas burn. What does your buddy get for mpg? loaded that is, the wind drag is the same with a travel trailer, unloaded these are different vehicles and not quite so comparable.