You are accounting for the weight of the camper and the people but have totally forgotten about the weight of clothes, food, cooking equipment and everything else that gets brought along. Also most camper weights provided by manufacturers leave out anything listed as an option (oven, A/C, awning, extra/larger battery, etc) along with the weight of water and propane. All of this stuff can easily add up to over 500 pounds.
Why the concentration on only an F150? Do you have registration or parking limits you are working within?
While I haven't done it in a couple years, the two times I went to Ford's website and did so I found it was possible to spec out for factory order an F250 that cost LESS then a comparably equipped F150.