The rear axle limit is probably more than just suspension, the axle tube, bearings, lug nuts, etc are all 1/2 ton rated. The only way to increase the rear to a 3/4 ton is to sell and buy a 3/4 ton. There really isn't a way to increase payload... It is what it is, that trailer and others with similar weight are too big for your truck.
Look for a smaller trailer. 6900 dry is a heavy camper, the one you listed is over 31' long.. I know Georgia is flatter than up here, but I tow a 29'3" TT that weighs about 8500 fully loaded with my 1 Ton SRW, essentially the same as a 3/4, and I wouldn't want to tow it with a 1/2 ton.
You will likely exceed
GCWR
GVWR
RAWR
If you want to know where you are at then load up the family, put anything in the truck you would carry on a camping trip, WDH, put bikes/etc in the bed, top off the fuel tank and hit the scales. Get a weight on FAW, RAW, and GVW. I would add 200lbs to the RAW and 100lbs to the FAW (increase of 300lbs on the GVW). Then subtract that number from 13,000. That is your MAX trailer weight. Then take the RAW and subtract it (with the added 200lbs, and WDH installed) from the RAWR. That is the max tongue weight you can have.
One other thing is be careful with published tongue weights, they are, in my experience, always low. My TT was published with a TW around 750lbs and I am not sure I have ever seen it below 1,000 with an empty trailer and it's been well over 1,300 with a full trailer.