5.3 is proving to be a solid motor but your at the upper limits ESPECIALLY when you get into high altitudes. The make or break deal will be how that burb is equipped for the rear axle. Ideally a 4.10 gear and a Semi Floating 14 bolt is what you want to find. (count the bolts holding the cover on). That's a sturdy axle with a good size ring and pinion that will stand up to the abuse of trailering.
In the glove box is a sticker with a bunch of codes on it. Those codes represent everything on that vehicle. They are called "RPO" codes. One of those codes will tell you the gear ratio. The higher the number the better. Like I said, 4.10 is what you want to find. 3.73 is pushing it.....really pushing it. anything less then 3.73 and I would say it's too much trailer for the tow vehicle. Your going to have no power and it will be very hard on the transmission.
Now that said you can change the gear ratio. It's going to set you back about $700. You might even hit the local junk yards if you have a 10 bolt. You might be able to pick up a 6lug sf 14 that will be a bolt on for about the same amount of money. Many Z71's had that axle and as long as the spring purch width is the same it will be a bolt on. The computer for the engine will also need to be reconfigured for the gear ratio. When they reprogram it they can revise the shift points to hold gear a little longer and shift a little firmer.
You also need to add a big transmission cooler. A Stack plate cooler is more efficent then a tube and fin design. If your a do it yourselfer Pep Boys stocks Hayden brand stack plate coolers. I have put these on my last three trucks and been very pleased with them.
How far are you from Denver? Here is a shop near Denver that could handle the gear change or axle swap.
www.offroaddesign.com Talk to Stephen and tell him "Grim" (My other internet name that he knows me by) sent ya. He might even be able to locate the axle for you If you have him do the work. He's a straight up guy and buisness owner. I have delt with him many times. He does gear and axles changes regularly.
Not sure if he has the equipment to reprogram the ECM. It's a 4x4 shop and most of the stuff he does is custom work on older vehicles so may not have the equipment but I bet he knows somebody that does.