Howdy and welcome to the ORG.
Just looked at the H&S website, I didn't know they made a tuner for the 6.0L.
It only mentions transmission tuning for the 6.4L Powerstroke, and 6.7L Cummins, not the 6.0L. Also, it appears to have 3 set power levels, and no real custom tuning from H&S.
Here's where most of the tuners fall short, no transmission tuning. The 5R110 is holding upto 500RWHP, as long as its tuned to support that type of powerband. Most of the tuners out there, H&S, Banks, Edge, etc. rely on the transmission relearn to adapt to the new power. These method is very hard on your transmission, and will break things if you change tunes often.
Do this once and get the right setup that has been proven by a number of drivers on the ORG. Get a ScangaugeII or Edge Insight CS/CTS to monitor your engine, with a miminum of the expandable EGT probe with the Edge. Monitor ECT/EOT and ensure your oil cooler is in good health, along with FICM voltage at start up and idle. The tuner you want is the SCT, because it is a direct plug to the OBII port, and you flash the truck. It does not have shift-on-the-fly capability, but it does tune the transmission. The stock files on the SF3 are ok, but to take full advantage of the tuning you need to run custom files.
Eric at Innovative Diesel, Matt at Gearhead Automotive, and Vivian at Quick Tricks, are the big 3 tuners I've seen here. I run tunes from Innovative, have haven't had an issue. For a non-studded truck, I would stick with the street level type tunes pushing 400RWHP. Make sure you tell the tuner the mods you've done to your truck when you email your trucks strategy to them. For studded, you're free to run anything.
Biggest reason I got the tuner was for fuel economy, with a stock hieght dually and 4.30 gears I see 13-15 mpg stock. On my street file its 14.5-16.5 mpg, and race is a little better than that when I can keep my foot out of it.
If you have any questions, don't hesitate to ask.
I was ready to buy a Banks Six-gun when I found this site, and I'm sure glad I didn't.