LE3105 is way too much hauler for a gas motor truck.
I have a LE3505, with 4' added, so LE3905 and my trailer weighs 11,600lbs totally empty.
With toys, water etc, I am 17,000lbs.
All of my weights are real life CAT scale weights, not brochure weights.
The SL series of the Warrior line will be more in line of what you are looking for.
You are reading your manual correctly. 3/4 ton trucks don't have as much towing capacity as folks think. Especially when you add a crew cab and diesel into the mix. I know you don't have a diesel, just giving an example.
Load range E tires should be used on the truck. IMO.
Bigger than stock tires effectively change your gear ratio. So, unless different gears were installed, your tow ratings are actually lower than you think.
Manufacture tow ratings are based on lots of variables, frame, brakes, axles, tires (size and load rating), motor HP and Torque etc. Any of these change, changes your tow ratings.