Tow ratings by vehicle manufacturers include all types of trailers. RVs are probably the most difficult type of trailer to tow due to the high profile (front and side) and therefore need the higher end of 12-15% tongue weight for stability.
Starting at 1220 pounds of payload, take off 400 for couple plus cat leaves 800 for loaded trailer tongue weight. As you can see, the dry TW of that Innsbruck exceeds that and any dry TW will likely go up by about 200 pounds when loaded wit camping gear. You will want to start with something closer to 6000 pounds dry with 600 pounds dry TW. And you will still likely exceed the truck's GVWR/payload as I have never witnessed a pickup truck pull into a campsite with an empty truck bed.