I towed a 4800 pound fiver with a V6 T-100 extra cab thousands of miles. Don't tell me the DC 5.7 is limited to 5000 pounds!!
Well I towed a D8 to alaska with my titan, but I can't tell you that's safe.
With roughly 900lbs of pin weight a full tank of fuel, a wife, some firewood and a cooler, you're overweight in a 2003 tundra access cab. You may have reloaded your trailer to limit the pin weight, but you either weren't safe, or you weren't legal. Those are two different things, but it's your insurance policy.
I own a 4000lb 5er that I towed (for thousands of miles) with my supercharged '03 access cab. I can honestly say that I would not put a larger trailer behind that chassis. Further I could not put any more trailer on the truck without exceeding the RAWR.
IF you're planning on towing legally, get a fifth wheel with less than 1000 lbs of pin weight (roughly a 5000 lb trailer) and leave yourself the 600 lbs of remaining GVWR for coolers, passengers, fuel, chains and firewood. OR buy a travel trailer.
If you're not concerned abour axle weights or GVWR then fill your boots and carry a camera to take a picture of the carnage and the scale fines.