I don't think any motor mods are worth the expense. You just don't get much bang for the buck. Not to mention voiding your warranty. However, regearing the differential(s) can make a big improvement. You most likely have 3:73 now. Going to 4:30 gears would impress you, and reports are that it doesn't hurt fuel mileage too much if at all.
Yes, on a 4x4 you have to do both axles which doubles the cost. It's not a DIY project either. Take it to a good gear shop and have it done. Runs around $750-$1000 PER axle.
But the point is, it is the only thing that will make a dramatic difference in towing power. Exhaust, intake, tuning, will only make small improvements in HP and when towing you won't feel a thing in the seat of your pants except for a thinner wallet. You will FEEL a gear change.
Those trucks weigh so much the extra 40hp over the older models is needed just to move its bulk. Those engines are very nice though, but there isn't much you can do to them to make it run nicer without dumping big bucks for little bang. Diesels allow MUCH more return on your cash than gas engines do.
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.