2012 Ram 1500 / Shasta Oasis 310K

You'd be overloading your truck by a significant margin.

Especially with half-tons, the trailer towing rating is a misleading and usually moot number. Other ratings, like axle weight ratings or your gross vehicle weight rating will likely be your limiting numbers. Your payload is the biggest concern. You state a payload capacity of 1322 lbs. Add in people, dog and supplies, and you're at 800 lbs, leaving 522 lbs.

What you're calling hitch weight is actually called tongue weight - the weight that the tongue of the trailer will put on your truck when it's connected. You actually do have to take into account the hitch weight - the weight of the ball you're going to insert into your receiver that will connect the truck to the trailer. This is usually at least 40-50 lbs. So that takes you down to 482 lbs that's available for the weight of the trailer tongue.

A properly loaded and balanced trailer should have a tongue weight of about 13% of trailer weight. So a camper with a dry weight of 5700 lbs should have a tongue weight of ~741 lbs. Once you add supplies, food and other gear in the trailer, you're probably adding another 100-200 lbs of tongue weight. That would put the tongue weight closer to 900 lbs.

So, you're ending up with under 500 lbs of available payload for an almost 900-lb tongue weight. You'll be exceeding your GVWR by several hundred pounds, and probably one or both of your axle ratings.

I know some people dismiss the ratings on the door tags and say, "tow what you're comfortable with." My personal experience showed that once I started flirting with the max ratings on my half-ton truck, stuff started breaking while we were on vacation. Forgetting about the inherent safety issues in doing that, you have the added fun of explaining to your family why they have to spend four days in Mitchell, SD instead of Yellowstone Park, waiting for the truck to be repaired.

I think you'll do ok.
edit:
You just need to make sure you get your hitch set up right. I also find it very beneficial to add air in the tires when towing, especially the rears. You will also need to keep the pan cargo to the barest minimum. All true for a lot of 1/2t trucks.