We did it 3 weeks ago with stiff mountaineering boots (I was using Trango Ices, my partner had Nepal EVOs). They made the snow travel extra-secure and were just fine for all of the rock climbing. I'm a big fan of trail runners, but if the snow traverse were firm and didn't have pre-kicked steps, it'd be very sketchy to use trail runners with crampons. That said, you might find success dropping over the S. Side there and avoiding the snow traverse, as per Steph Abegg's report.
For what it's worth, my partner brought rock shoes and never bothered to put them on though they might have been nice in some of the slabby, loose, sandy class 4 on Torment. We did carry crampons (mine were the steel Kahtoola KTS's), and I'd say they're mandatory with all of the glacier travel. Aluminum would have worked for our trip.
[added] In retrospect, it was a huge time-saver to not bring other footwear. No faffin' around changing shoes, just a bunch of long simul pitches. Bring extra double+ length slings to extend simul blocks, since most of the pro consists of large slung horns.