My feeling is that the F250 will be marginal with that engine. The reason I say that is because my wife has a 2007 Mountaineer with the 5.4L V8 and I would not want to tow that much with it. The F250 will certainly carry the tongue weight of the trailer and have plenty of carrying capacity but I think the 5.4 engine might be a bit weak. Of course, a lot depends on the rear end ratio and other factors. The truck may work fine and then again it may not. Sorry I cannot say for sure.
Were it me, I would keep looking - possibly for a V10 or even a 7.3 diesel like mine. They made the 7.3 up until early 2003 when they switched to the 6.0L in mid year, which you probably should avoid from all that I have read.
Barney