I have been pulling 5800 lbs with a Suburban for three plus years. That truck's going to feel 6800 pounds. I don't know what your max weight rating is but, I'd recommend towing with empty tanks. I like the Chevrolets. When pulling a hill don't let the transmission decide what gear it wants. You decide. Let the RPMs stay in a comfortable RPM range. When pulling Black Mountain I pull it down and keep the RPMs over 3000 (about 3200) and that puts me around 47 MPH up the mountain with the AC on. I added Sumo Supersprings to the rear end and an additional transmission cooler. Don't tow in overdrive. Don't allow the transmission to hunt. I'm super anal about servicing. A 2500 would always do a better job but, I love my old 1500 Suburban and it gets the job done. We need the extra air conditioned space because we take our dogs with us. There's always better options if you have the cash.. But, for my 27 foot, 5800lb (dry) travel trailer the 1500 Chevy Suburban is adequate. If and when I trade I'll be looking for a big SUV, maybe GMC with the 6.0/6.2. The chassis handles the weight fine. It's super important to get that WD hitch set up right.
My max weight is 7200. I wouldn't want to commit to pulling a 6800 dry weight trailer around. That gives you very little, if any wiggle room on weight. Will it do it? Yep, I think it will. But, it's at the top end (or over). There are some people who store their big trailer near mine that I'm sure is at their weight limit. But, they pull it with a 1500 Suburban 5.3 all the time.
I tend to baby my transmission. I'm not in a hurry. I don't allow it to hunt for gears. I get a running start on hills where possible and let the speed slowly bleed down on the way up, etc.. The 5.3 isn't the problem. The GM 4.8/5.3/6.0 is a good design and don't mind RPMs. The powerband on the 5.3 requires RPMs and that's the way it was designed. The 4L60e transmission is a good transmission but, it needs to be serviced. I pulled the pan and replaced the fluid and filter a couple of years ago. I do a drain and fill on the Dextron 6 a couple of times a year. The fluid stays red and fresh all the time. I also service the differential every couple of years (it was rebuilt in 2017).