I don't really think you will notice much of a difference in fuel economy with different tires. If you are comfortable with the towing characteristics of your truck with P rated car tires and don't want the extra weight of the LT tires, stay with the P rating.
I have E rated tires (WAY overkill, but came standard with the Rock Warrior package) on my Tundra and I have no flex or sway. I wouldn't go back to anything less than a D rated tire just because it really makes the towing platform much for solid.
For towing, the LT tires are better. They are more rigid and have less sidewall flex. That leads to lower tire temperatures and less chance for a tire failure on your tow vehicle. That also leads to less swaying or bouncing feel while towing.