Reposting as the other thread got closed..
Having towed with multiple different rigs, Ford 5.4 With 3.73 rear end a 4-speed, GM 6.0 3.73 and 6-Speed, Ford F250 6.4 Diesel I have to ask a few questions. The interesting part on the hills was it from lack of power? If so I honestly don't believe that going to a 1500HD with the 6.0 is going to be a ground breaking difference. Yes it will be up to the task, and it was designed for that purpose. But your also getting a heavier truck, larger axles, more springs more weight so the power gain is limited with a large load behind it.
If it's a suspension, breaking, stopping perspective on hills that concerns you then yes the 1500HD is better suited to do that job and do it better as the 1500HD is essentially a 2500 Light Duty.
The 6.0 for that model year range puts down 300HP and 360TQ, the 5.3 for 2004 put down 295HP and 330TQ. The gearing will help but it's not going to be anything significant. In fact I have known guys that have "upraded" to 3/4-tons of that generation that have been disappointed because pulling power wise there is no difference, what your gaining is suspension, braking and control of the trailer.
Just FYI it's really worth it to consider what it is your concerned with and wanting to gain.
2009 Ford F350 V10 4.10 FX4 Crew Cab SRW, Timbrens, Leer Topper