I downloaded the development version of OpenSCAD which includes its own Customizer interface.  I tweaked the OpenSCAD code to allow my model to work with this customizer.  This is now working (Yay!), so I replaced the earlier uploaded model with this version (which uses Mills  [1/1000 of an inch] rather than fractional inches so the sliders in the Customizer render properly).

Hopefully that will make the integration with Thingiverse's Customizer painless (assuming they ever fix it).