To tow a 4800lb trailer with only 400lbs tongue weight, NO, you do not need a weight-distributing hitch on an F250.
Be aware that is the EMPTY weight of the trailer. Add two tanks of propane, batteries, air conditioner, television, a tank of water, your clothes, food, bedding, pots, pans, silverware, dishes, toys, games... It will get much heavier.
When the trailer is all loaded up, you MAY need a weight-distributing hitch. This will depend on the ratings of the truck's receiver vs. the actual tongue weight of the trailer.
You may WANT a weight-distributing hitch, even if you don't NEED one, for other reasons. Many weight-distributing hitches also incorporate sway control that is much better than the old-fashioned friction bars. Even if you don't feel you need weight-distribution or sway control, it doesn't hurt to have them.