I'm going to use my truck as an example and you can make your own decision on that trailer. I have an '11 QC 4x4 3.92 gear and LT275/70R17 tires. The GVWR is 6700, GCWR is 15,500. Payload is listed as 1610 and max trailer wight is listed as 10,050.
With 265lbs of me, 32 gallons of gas, and maybe a little over 100lbs of crap in the truck it weighs 6080(3420 front, 2660 rear). With my trailer hooked up my truck went up to 6760(3420 front, 3340 rear). The trailer was 6140 on the axles. It added up to 12,900 total weight.
So lets take a look at the numbers. The trailer(30' overall TT) at 6820 is well under the max trailer rating. However at 6760 I am over my GVWR. Legally probably a big deal. Ability wise it's not. The GAWR on the rear is 3900 so I'm still under that. With a trailer a couple thousand heavier you should have about another 200 on the truck.
On to my story of hooking up the camper. When I got my truck I moved my ball 1 hole higher(quit snickering) and hooked the weight distributing hitch the same as my last truck. That thing was all over the place when I pulled it. I adjusted things 2 or 3 more times until I got it set right. Now it pulls like a dream. I pulled it about 175 miles twice in the last 2 weeks and didn't use my sway control. It still pulled fine. I couldn't even tell it wasn't on.
Setting the brake controller is next thing. This is an incredibly important part of pulling a trailer that heavy. As far as braking the trailer will stop itself when set right. Go through the steps to do this. When it's been sitting a while check the settings again. You'll probably need to drag the brakes a bit to clean the drums and shoes.
I hope this helps you. I didn't want to just tell you that trailer will or won't be fine. I do think the truck is capable of pulling it if everything is set correctly. If you don't have much experience pulling trailers I would be a little hesitant though. Don't get in a hurry, look plenty far ahead, and pay attention to everything. The big issue would be a potential legal one if you're over your weight ratings. Good luck.