Lookin' good already no doubt
FYI - I'm runnin the same Sturmey Archer hubs and the three speed is holdin' up really well to the SBP Deluxe HD Shifter Kit II, although you may find the axle nuts difficult to tighten adequately w/o stripping them to prevent movement in the dropouts. The problem is the flats on the axle (to prevent rotation) effectively reduces the threaded area by about 1/2 & there's just not enough surface area to hold under
hard acceleration. It's not a fatal problem, just annoying when ya drop a gear and punch it to pass some spandexer and promptly derail the chain as the rear sprocket is a lil crooked now
While my bike came equipped with dropout tensioners, you may wish to consider something like this as a solution:
Bikeman: MKS Track Dropout Tensioners, for 5mm Dropouts (but I dunno yer axle shaft & dropout sizes so better chk them bfore ordering).
I'd also advise usin' a drop o'loctite on the Sturmey's cable tension adjusters - I spaced that and paid the price of five spokes ripped outa my front wheel

Fortunately I was puttin' along at about a walking speed at the time lol
That's such a sweet lookin' ride already man - I can't wait to see where ya go from here
Oh right - oddly, the Sturmey 3sp seems geared a lil tall with the shiftkit (the shiftkit gearing was with a 21sp bike in mind), I'm running the 10T (standard, comes w/kit) jackshaft to crank sprocket (lil one on right side) with the 48T chainring (kit comes w/the 44T):
Chainring - Freewheel - 48 Tooth w/chain guard and that seems about perfect for my ride... which is a bit of a heavy bike *shrug*