Manufacturer tow ratings seem to be a mystery to most. I believe that the tow ratings are a function of cooling.
There are some bigger engines that have little power but with big tow numbers. Then there are some small engines with big power but with little tow numbers.
I don't believe that most makers rate towing by power. I believe torque has a lot to do with an engine working less, and producing less heat, which equals bigger tow numbers.
The axle ratings help as a torque multiplier in allowing these engines to work less, and produce less heat. Hense why a short gear/higher ratio axle will have a higher tow rating.
A smaller displacement engine seems to produce higher heat as well. Despite the power rating, the engine size seems to be a function of heat.
Ford is a good example since that's what I'm familiar with.
The 4.6L and 5.0L engines are small displacement and have the same tow ratings as long as the same transmissions and same axle ratios are used.
The 5.4L with the 3.55 axle ratio has a higher tow rating than the 4.6L/5.0L with 3.55 or 3.73 axle ratios despite it's lower power output. Even though the 5.4L technically has higher torque at lower rpms than the 5.0L, the 5.0L with the 3.73 axle torque multiplier more than exceeds the torque of the 5.4L with the 3.55 axle. None the less, the 5.4L still has the higher tow rating due to it's physical size and ability to better cool.
Your true tow ratings is likely lower as the tow rating is calculated from your GCWR. The GCWR is what your truck will weigh loaded and your trailer with weigh loaded. The tow rating is calculated with the lightest base model with only a 150lbs driver. Add more people, weight, or gear to the truck, it all has to be subtracted off of your tow rating.
Your GCWR is 12,200 lbs and your tow rating is 6700 lbs, so they are saying for that tow rating, your truck must only weigh 5500 lbs. Hek my truck from the factory weighs 5768 lbs. but I think comparable GM products are about 200 lbs lighter.
We add about 800 lbs to our truck with people and gear so that would drop your true tow rating down to 5900 lbs. That will likely not be enough tow rating for those trailers unless you can keep it down to that low tow rating along with only driving on level roads, no hills or mountains, in normal heat. Drive in hills, mountain, in high heat, and if those trailers are over the tow ratings frontal surface area, you true tow rating will even be lower. That's why many recommend only towing 80% of your manufacturers tow rating, to allow for those issues as well. These once again, will cause your engine to work harder and to make more heat. Once again coming back to the cooling of the engine.
The 3.08 ratio was not meant to tow with. In fact I believe GM doesn't even allow it in the standard tow package, requiring a minimum of the 3.42 axle.
I'd stick to a trailer that's 5000 lbs loaded with that low output 5.3L engine and that fuel efficiency 3.08 axle. The combo isn't meant to tow anything but lighter trailers. Even my midsize 2008 4.6L V8 Sport Trac had better gearing allowing a 7000 lbs tow rating.
If you want a bigger trailer, you could always trade it in. Sucks that's it's new and doesn't have the ability to do what you need, but you'll make sure next time. I've gone through that myself with my previous 2004 XE Titan that only had a 6500 lbs tow rating. At that time, I ended up going with a Pop Up anyhow since I couldn't find any used travel trailer in good shape.
