Preliminary report is in. I installed a wheel with a Nexus 8-speed ("Red Band") hub and the noise is absent. The hub is in its lowest gear, as I didn't want to commit to the re-fitting an 8-speed shifter, but stomp as I would, the noise would not appear. With the Alfine 11, the noise was present in all gears.
So I'm going to finalize the conversion from Verge S11i to Verge S8i. If the 11-speed hub can be repaired, I can go back at my leisure.
FWIW, I'm not complaining about the Alfine hub; it has up until now been dead-reliable, and it has 10,600 miles on it.

When I took the chainring off to clean it, I noticed some pretty severe wear on the teeth. The chainring is rather thin, IMO, for a bike that uses a single-speed chain. I was sort of hoping the noise was somehow due to the chain sliding from side-to-side. No such luck.
I'll post findings of anything I notice when I open the Alfine.