The stock Sunline rims are NOT rated for the pressure level of Class E tires so the carrying capacity is not increased by putting a higher Class tire on it. You do get a heavier built tire, though, that may be more durable. Heavier built usually also means: produces more heat and offers more resistance when rolling.
A wider rim for a wider tire lets the tire wear and handle better. A rim too narrow concentrates the weight carrying on the center of the tread. Myself, I'm staying with the standard size and range which are totally adequate for anything I'm going to do. More important than the size and Class, which are completely adequate, is the quality of the tire.
2007 Ford F150 FX4 Supercab 4x4, 5.4 FFV engine, 3.73LS.