Here's a link to 2005 Ford Towing Guide.
Your truck with factory sized tires and 3.73 has a towing limit of 9500# (if 4X2). The oversize tires would reduce the axle ratio to about 3.31. They don't list that combination, but following the reduction for 3.73 to 3.55, the limit would be about 7500#. The GCWR also goes down dramatically with axle ratio.
The camper with propane and battery(s) will be probably over 7100# before you add anything else. With gear and bike, you'll add approx. 1700 - 1800#, for a loaded weight of 8800 - 8900#. The bikes in the garage will reduce the tongue weight, but you ideally want about 13% tongue weight for good towing manners...that'd be about 1150#.
That's probably higher than the payload available on your truck depending on the number of passengers and gear. To know for sure, a weight of your truck loaded with fuel, passsengers and gear subtracted from the truck's GVWR will tell you how much payload capacity and GCWR you have available. Just ball-parking the numbers say that you'd be at the max if not over the truck's limits.