Yes you HAVE to use a WD hitch to tow 7000# with the GM hitch per the sticker. You must follow the rating on the hitch so you have no choice, you'll need to replace the hitch or get a W/D system for 7000#.
I am a big fan of W/D hitches. I have one for my EQ trailer and one for my RV trailer. When the EQ trailer is empty, the tongue weight is reduced and the WD system is just relaxed. It doesn't hurt anything to tow with it this way. That said, on long trips with an empty trailer I'll just use a regular ball hitch that is already in the receiver to avoid messing with the chains. I use an F350 to tow 7000# or less and still use the WD system since they make the whole towing experience better. I often turn my trailer until it almost hits the truck to back it in to tight places. No problems with the WD hitch unless you set it up wrong.
I have also used timbren overloads on my last truck, a 1998 chevy half ton with leaf springs and they were excellent with no moving parts, no effect on unloaded ride, and they were cheap. I don't recommend air bags unless you have no other choice. I don't recommend air springs at all, they load the shock mounts with tongue weight which were never designed to carry that load. Yes on the LT tires, even my half ton always got D rated LT tires. Much better but whatever you have on the burban should be rated highly enough to carry the rated load of the truck.
Cheapest way to legally tow the load you want, right now, is with a W/D hitch system. The last one I bought was a Robyn rated for 10,000# and 1000# tongue weight for under 250$. If you really hate WD hitches then your second cheapest route is to replace the hitch with a special stout hitch rated for over 7000# on a non-WD hitch. I can tell you that my OEM F350 hitch is not rated for 7000# without the WD system so you might need a pretty heavy duty aftermarket receiver hitch.
Your second issue is that you think you need to beef up the suspension. Why? Try towing within the ratings and see how the burban squats. 7000# trailer means about 700# tongue weight and that's like 3 or 4 fat chicks. Don't stuff the back of the burb with gear and it shouldn't bottom out much. If you need to beef it up see what timbren has for you.