I've posted some lengthy stuff about towing with these vehicles. But it's been a while, you'd have to dig for it.
Look in your glove compartment door for the sticker listing your vehicle codes. GT4 is 3.73 gears. Z82 is the factory tow package. That will be further evidenced by having the 'tow' button in the end of your gear shift lever. This changes the computer set points for engine / trans shift points. Pressing it will activate the system and a yellow indicator light will show in the instrument cluster. Yours ought to look like my '02 -
If you are towing a heavy load - and a full vehicle and a 4500 trailer IS a heavy load in a 1/2-ton Sub - you'll want to take it easy and keep the selector in 3rd for anything but cruising down a flat highway at speed.
The high-mileage 4L60E is your greatest concern. Baby it. Make sure you have a trans cooler installed if you DON'T have the towing package. Might even consider a deeper trans pan with greater fluid capacity. Change your trans filter if you haven't. Make sure your engine cooling system is serviced and in top shape, as part of the trans cooling is plumbed thru the radiator tank, before then looping thru the external cooler (which is mounted low in front of the radiator, just below the bow tie as you look in the grill - it's not a great place for it, but it's the least worst). Do everything you can to ensure your engine and trans are cooling as efficiently as they can. Get a trans temp gauge or computer-reading tool that will show you. And keep your eye on it. Slow down or take a break when the trans heats up. Or make the family sweat and use your interior heaters for all they are worth, to bleed heat out of the drivetrain.
We've got an '02 Sub and '05 Tahoe, Z-71s, we've towed a good bit in a variety of conditions with both. We have the same drivetrain config as you. My Sub has 121k mi on it, her Tahoe just crossed 150k. Hers is practically 'new' in performance and condition, had it from mile 6. Mine was a bit shabby Mom-taxi when I bought it used at 114k, it's trans is a little sloppy. I baby it. Just put a gallon of 'hi mileage' trans fluid in it, as an experiment, seemed to help. That's not really good news. But I'm only driving it about 5500mi/yr. So it will do.
The 5.3L is of no concern, that thing will pull hard all day.
Nothing to be particularly worried about, just take reasonable care and maintenance.
eta just looked and LMC Truck has the OEM trans cooler for $130 and the two modified lines needed to hook it up are ~$30ea. Call it ~$200 to add a factory cooler, if you don't have the factory tow package.