Op says his door sticker states just over 2200 lbs of capacity. Old biscuit wrote the 5ers wet pin is/should be 3200 lbs. My guess the truck itself weighs around (don't recall seeing OP's set up) 8500. If that's the case, legally only has 1500 lb avail for extra being it's a 10,000 GVW truck. So assume for a moment we use OB's numbers along w/ an assumed truck weight. 8500 + 3200 = 11,700 which according to my fingers and toes puts him over the trucks GVW by 1,700 lbs. Shoot, that's actually over a 350's GVW of 11,500 by 200#.
IMO OP, yes your truck can pull it, but legally if you ever get into a wreck w/ another vehicle, some sue happy att'y will go after you tooth and nail. May have to either look at a different RV or swallow hard and get a 1 ton truck too.
Let me give you a personal experience. Bought a '15 F350 SRW 6.7,4x4,cc,sb, GVW of 11,500. At that time we had a '06 Cedar Creek w/ a max GVW of 13,800, wet pin of 2100. Prior truck was a '03 F250 6.0,cc,4x4,sb. Yes the '03 is in same boat your are looking into, but that's what we had. Anyways' the '15 was no prob w/ the Creek. Year later, we decide to upgrade RV to one in sig. Max wt is 16,000, dry is 14,000, wet pin is 3700. Yes the '15 pulled the '17 RV ok, but back in same boat. Over on most weights by a little or several hundred lbs. In '18 we decided to go to FL in Aug for vaykay. Biggest overweight on the '15 was tires and knowing all the extra heat heading to FL and back was gnawing at me like crazy (basically a gut feeling in stomach that wouldn't go away). As much as it pained me to do so, traded the '15 for a '17 F350 DRW. Towing is a dream, wife feels more at ease w/ it when she relives me at the wheel and if ya have a rear wheel go flat at 65-70 mph, at least ya got another one to get you off the road. Yes the DRW absolutely stinks in the snow and is a bit more to consider at drive thrus and fuel pumps due to the hips. BUT from a heavy RV towing aspect, wouldn't go back to a SRW. Would I go back to a SRW? Yes w/ a lighter trailer. Yes you can put on/add larger tires, suspension enhancements, but bottom line is what's legal for your truck? Today's SRW's do have higher GVW's than those of just a few years ago, but I'd personally be leary of putting anything much over 15000 on a SRW 1 ton and DEFINITELY NOT a 3/4 ton.