That Toyota cannot "handle" 6500 pounds. It may be able to tow it, though. But it can only "handle" maybe 1000 pounds...as in total payload capacity. Subtract from that all the weight of passengers, hitch, and all the junk you carry. Then see if the tongue weight of the LOADED FOR CAMPING...trailer is less than the payload capacity you have left.
I kinda doubt it. Go for a bigger truck.
Incidentally your numbers are faulty. Not a 4963 pound pin weight on a 32 foot trailer. That may be DRY weight. But dry weight means nothing when calculating tow capacity. Use the GVWR rating of the trailer. That is a closer estimate.