It's not about tow ratings, the GVWR of the truck comes into play also...My TT is 8500 lbs loaded. Tongue weight is right about 1200 lbs. The weight of the Suburban with me a tank of gas is 6700 lbs, and the max weight is 8600 lbs. With me , wife, my two kids and a friend's kid along with the tongue weight of the trailer, the truck weighed in right at 8600 lbs. Yes I have gone camping with a loaded cooler and firewood etc and probably had the truck at 9000+ lbs on some trips.
The average family adds 1500 lbs of "stuff" to the dry weight of their trailer, and we are right on that mark. With a 37' trailer and 6 kids I would say you easily add 2000 lbs., so that makes the estimated loaded weight of the trailer at 12,000 lbs. 12% on the tongue means 1,440 tongue weight.
So on the 8.1 and compared to my set up you will be adding an extra 240 lbs of tongue weight plus 3 kids at 100 lbs each, plus all the "necessities" that kids bring, I would guess you'll be hitting the road about 9600 lbs or about 1,000 lbs overweight. The 4.10 gears give you a higher tow rating and GCWR but the GVWR is the same at 8600 lbs.
IMO the truck will handle it but you will be on the bleeding edge with that trailer....the burb has a comparatively short wheelbase for a tow vehicle, and I would strongly advise a Hensley or ProPride 4-bar linkage hitch to keep that trailer under control. At 37' long, that's a lot of sail area to catch the wind, that rig could become a handful in a hurry. Note the pic in my signature, that TT is only 32' and I use a Hensley hitch to keep it well behaved. The burb looks like a toy relative to the trailer, and your trailer is 6' longer than that....
A much better tow vehicle would a 350 or 3500 crew cab long bed truck, you can get a GM with the 8.1 or a Ford with the V10. Yes, you lose the 3rd row, but you gain and additional 36" of wheelbase and a whole lot of GVWR carrying capacity to control that big honkin' trailer.
I love my burb but I think 10k loaded is about the reasonable limit for it. If you owned the burb I would be telling you to look for a smaller trailer, but since you own the trailer I'm telling you to look for a larger truck.
Note that the 8.1 is not the limiting factor, you could pull a house off its foundation with this engine. The limiting factors for the burb specifically are the shorter wheelbase and 8600 lb GVWR
