We just went through a search for the "right" Suburban, and I can feel your pain. Seems that 90% of the Suburban/YXLs out there came with the 3.08 gears and w/o the K5L towing package. I had to look long and hard for 3.42 gears and the cooling equipment -- probably about 30 trucks between 2009 and 2012 before I found 1 that was set up that way I wanted.
Here's what I would do in your situation. Disclaimer: I've been towing heavy things since I was 14 with an agricultural exemption license, giving me about 24 yrs of towing experience. So I'm very comfortable with a trailer behind me, and feel that I can pretty well analyze what is going on and what needs to be done. I also understand vehicles, transmissions, etc. pretty well, and understand where the weak points and problems arise. If your approach and intent is to hook up a trailer, put 'er in D and push on the right pedal until you reach the speed limit (or die trying), please stop reading now ...
The "tow rating" on the 3.08 trucks is frustratingly low, to be sure. But aside from the cooling equipment and the lower gears, the rest of the truck is the same as the higher tow rated version -- same brakes, same frame, same transmission, same engine, etc. So, the same "safety" stuff. The coolers and lower gears will help with equipment longevity and performance, but it isn't like the frame or brakes on a 3.08 truck are going to spontaneously combust or disintegrate under a load that would be fine behind a 3.42 truck. You are just at a higher risk of overheating your transmission and having to run in lower gears. Furthermore, the difference b/t the 3.08 and 3.42 isn't mind-boggling; it is a couple hundred RPMs. In reality, you're going to have to pull in 4th or 5th on flat ground in either setup; the 5.3 wont have the power to keep you in top gear unless you're going downhill. Either will do better than our prior 'Burb w/ 4.10 gears and the 4 spd transmission, of that I am confident.
So for moderate recreational use (read: not taking cross-country trips or towing thousands of miles at a time), I would buy a reasonable trailer, take a careful approach on the road, get a good sway/equalizing hitch (I like our EQ 4-way), and see how things work out with your current rig. The biggest thing is, you'll have to track your transmission temps closely -- fortunately, your truck has a trans temp readout in the DIC -- and make sure your transmission isn't hunting too much and all of that. If your tranny temps aren't staying down, install the transmission cooler. Personally, I wouldn't spend money to change out the 3.08s; just drive a gear lower with the wonderful 6 spd, and you'll be fine there.