It looks like for 2002 the Tundra was listed as having payload anywhere from 1366 to 1938 depending on trim level and options:
Trim Level Curb Wt. GVWR Payload
SR5 Regular Cab 4-speed automatic 4262 6200 1938
SR5 Access Cab 4-speed automatic 4518 6030 1512
LTD Access Cab 4-speed automatic 4644 6010 1366
It also looks like max tongue weight is just over 700lbs:
Trim Level Max Tongue Weight
V8 Reg. Cab 4WD 4-ECT 720
V8 Access Cab 2WD 4-ECT 720
V8 Access Cab 4WD 4-ECT 710
Max tongue weight will be a limiting factor regardless of how much additional payload you actually have. It looks like the max tow capacity has been calculated assuming a tongue weight of 10% which is the minimum recommended. Assuming you can limit the GVW of the trailer to 5500 and your tongue weight % to no more than 13% you'd be right at the max tongue weight without any margins to account for changes that occur while traveling such as tanks full/empty, re-distributed or added cargo along the way, etc.
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TT: 2019 ORV Timber Ridge 23DBS, Blue Ox SwayPro 15K/1500
TV: 2019 Ford F-250 XLT SuperCab LB, 6.2L, 4.30/e-locker, 4x4, 164" WB, RoadActive Suspension, 3216 payload
TV: 2014 RAM 1500 Big Horn CC (Traded in)
TT: 2015 Jay Flight SLX 195RB Baja Edition, Andersen No-Sway (Traded in)
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