Basically, wear whatever you're used to (trail runners). Nothing remarkably different from the east with respect to footwear on the 4-pass loop. Easier terrain footwear-wise than hiking, say, on a lot of the AT.
There are some spots where there is some talus and scree where a decent, solid sole is nice though. Not too much to warrant boots or anything silly like that. There are a couple of feet-wet creek crossings, maybe only one in September. "waterproof" boots/shoes would not help though. Too deep. If you want dry feet, just take shoes off, cross, put 'em back on. The crossing I refer to is in Fravert basin, about half way around the loop. It has a nice sandy bottom, no footwear required to cross.
"Talus and scree" are nicely define in Pmags article here:
https://pmags.com/talus-vs-scree-what-is-the-difference
And agree, no extra traction, though believe it or not, there might be some old leftover snow on the northeast side of Trailrider pass, even in early September. Epic snow coverage in those parts this year. I'll be hiking it soon, and I'll report in on conditions.