Update: Je suis un imbécile.
Rounding a sharp corner on my commute today, the drivetrain 'click' became much more pronounced. I stopped, determined to track this down once and for all.
Checked that the rear wheel was centred.
Checked that the freewheel was intact and secure.
Checked that the chainring had no play.
Checked that the chain was running smooth with no stiff links.
Grabbed the crank arm, pulled and pushed on it as hard as I could... and there was play. Not much, but enough.
https://imgur.com/WN8C5tr
When I got to work it was a two-minute fix with a 10mm and 1.5mm Allen, first checking that the axle was tight as it should be on each side, and then using the collar to correctly set the preload.
An apology is winging its way to White industries.
I'll say this though - it took a lot more force than I'd usually expect to detect side-to-side play in the axle on this setup. There is no up-down play of any kind at all, and now that the axle is snugged up it's as buttery-smooth as I remember it.
Let this thread remain as an aid to others, so they might not waste the time and energy that I did!