Just for the record, I did not say you needed a diesel. Although your dad would love the power. But you don't need that to tow 6k#.
I owned an '04 F150, 5.4L, auto, 3:73 gears, and pulled a 5,200# travel trailer with it, through the mountain passes of the Sierra. It did a good job, but yes, as you said it was working hard. How you drive it makes a difference. You need to let the 5.4L run at high RPMs. Don't be afraid to shift down and run it a 4k RPM. It's designed for that.
I traded up because I wanted diesel, not because my old truck could not do the job. It would have been financially better to keep the old truck and regear it. So it was a choice, not a need. I also wanted a better hunting truck, so that played into my decision too.
So, if dad likes the truck and wants to keep it, regear it. If he wants another truck, consider diesel.
Bill