If you've got the 3.08 then your factory tow rating is far lower than if you had the 3.42 gears. Don't have it handy but I thing in a 4wd Tahoe it's 5700 lbs with the 3.08 and 8200 with 3.42. Plus as I recall, the vehicles made with the 3.08 didn't have the factory towing package. When ordering the towing package option, you automatically got the 3.42 gears plus trans cooler and other upgrades. All Tahoes came with the class III receiver, that by itself doesn't indicate the factory towing package.
FWIW I'm considering swapping out the 3.42 gears in my Denali for 3.73 for a little more pulling power. When pulling my 6500lb race car trailer I can't ever use 6th gear, and can only use 5th on reasonably flat roads because the engine is only turning 2200 rpm at 65, far far off the torque curve. I'd like to get 5th gear rpms up so I can be turning 2600-2700 at 70 in 5th. (Of course that'll mean I'll be turning a few hundred rpm more in 6th when not towing, but since the absolute best I ever get out of the Denali is 19mpg at 80 on the highway, I'm ok if it drops to 18). I've checked with a couple of local respected offroad shops in my area and they've quoted $1000-1200 to do the swap for both ends. Said I won't need new bearings on a 30,000 mile old truck, they rarely go bad before 150,000 miles if fluids are changed regularly.
But remember to factor in a software change as well. Once you change gears you must reflash the ECU to compensate for the gear change. Otherwise your speedo will be off by more than 20%, your ECU won't react properly to throttle loads, and you could mess up all your shift points. Good thing is that the folks at Blackbear can probably handle that easily for you, and make a few tweaks at the same time.
