I have experience with the Nexus 7 and Sturmey Archer AW-3 (3 speed) hubs.
I question why you even bothered building a Crosscheck which is an excellant frame with an internal hub? You really need an 8 or 9 speed with a tripple chainring. If the Sram Spectro 7 wasn't low enough, I highly doubt you'll get lower with the Sturmey Archer 3 speed unless you install a small chainring and remove the sprocket for a larger one with more teeth.
I was riding my 7 Speed Bianchi Milano last night and the friction loss is so real, the performance loss especially in the winter compared to my Univega street bike (7 speed) with 700 cc wheels. The hub weights a ton and it feels like I have 2 heavy water bottles attached to the wheels. It's not a bad hub if you ride less than 7 miles with no hills. It's almost bomb proof in the snow.
I actually like the Sturmey Archer AW-3 because (in some ways) unlike the 7 Speeds from Shimano, it actually has a "Direct Drive" that does not suffer the same performance loss. You're right about the weight of the Sturmy being lighter does make a difference and you can certainly feel it. You'll spend 95% of your time in 2nd gear (direct drive) but 3rd gear has horrible friction losses and 1st gear is just slightly better. Still. If you can set up a bike (With an AW-3) that has a low second gear, then 1st gear just might be enough for those hills. You'll compromise 3rd gear but then again you were not intending to go fast with this bike were you?
I still think you need to forget the 3 and 7 speed and just build a 8 speed (cassette) with 3 chainrings. In you're case, you'll never be satisfied with the friction loses of the AW-3. I don't mind but then again, I only commute 10 blocks to the train station with no hills. Consider youself lucky you didn't spend $800 dollars for a 12 speed Roaloff (sp) hub that weights MORE!