Unlike ordinary bike tubes, Michelin ProTek tubes have a square-ish cross section:

So, having a defunct Michelin ProTek tube in hand, I cut, cleaned, and dissected a few inches, then punched out a set of (flat!) washers for the Badger Propel air fittings:

As with the cork version, they fit fine:

The picture is slightly fuzzy, because zooming a Pixel photo doesn’t magically create any new mmmm pixels.
I tested the washer with 45 psi air (the recommended maximum) and it holds the pressure fine. Better than a fouled ProTek valve, anyway.
Flushed with success, I preemptively replaced both OEM cork washers, an action which will surely come back to haunt me.