Your math for understanding the payload and tounge weight of the trailer is correct. If you did buy the trailer with the 7K+ GVWR, it doesn't necessarily mean that when you load it up for camping, you'll have that weight. For a weekend trip, you will probably load 800 lbs. You may not have a .13 ratio for tongue weight (although that is centered in the range of good ratios). Of course, the best way to find out all these weight numbers accurately is to visit a scale.
Beyond that there is good proof that hauling real close to maximum combination can stress the towing vehicle. It will be about how you feel about your Toyota and what the experience of towing seems to be. I've never towed with a Toyota but have with a full frame Explorer and my old Mazda truck. The Explorer was similar (better?) than a half ton truck. The Mazda could hardly pull a tent camper.
Good luck with whatever you get. Family RV'ing builds memories that last a lifetime!
