I don't have a 1500 so I can't speak specifically if the plug is under the dash but, on alot of trucks the plug is there. You''ll just need the adapter from the brake controller to that plug.
7100lbs is the max trailer weight that the truck is rated for by GMC. Your trailer is 5200lbs dry. Everyone's loaded weight is different but, a rule or thumb is 1000lbs some are less and some are more.
You'll need to watch your payload, check out the sticker on the door.
What type of hitch do you have? Brand, size? I think that your truck should tow this but, may struggle a little on some hills. If your hitch is setup properly, the truck should handle it fine.
If it's not setup properly, it could be a good handful for that truck.
There are many brands of weight distribution hitches, and they aren't all created equal.
I think the truck will do it, but only you will know for sure as comfort levels differ as well.
You can always set it all up and take it to a CAT scale too and see how you measure up. This will help with your WD hitch setup as well.
2012 Keystone Sprinter
Still sticking around for the good people and advice.