You can get nice All Terrain tires that can increase your capacity. Don't just look at the load rating, look at the actual tire rating.
Example, most "E" rated tires will have a rating of 2900 - 3200lbs per tire. Falken makes an "E" rated AT tire that rates at 3900lbs per tire, so going with this tire would increase your tow rating.
Be sure to match the rim rating to the tire, but a lot of aftermarket rim's will be rated up to the "E" tire ratings or slightly beyond. If you are looking for "G" rated tires then your rim selection is reduced dramatically.
As others mentioned, if you increase the diameter of the tire you change the gear ratio's of the truck, but sometimes that can be beneficial. Again as an example, my previous truck with stock tires was a diesel, with stock tires would turn 1900 rpm at 62mph. 1900 rpm was that trucks strongest pulling and best mpg point. I put larger tires that did not require lifting the truck and was able to get to 1900 rpm at 70mph. This change didn't impact my towing experience as the motor still had enough grunt to pull my various trailers.
I guess my point, is there is truth to the advice already given but also there is other options to consider.
Good luck in your quest!
2012 Epic Voltage 3905
2013 High Output Dodge Cummins