Fwiw, I see a FOUC on that page in Firefox but not in Vivaldi (my default browser, which is Chromium-based). Even in Firefox, it only happens the first time I select a style - presumably while the file is fetched and parsed.
Is inlining the theme-specific CSS in the document head an option here? That should fix the FOUC, and given the small size of the css files (they're ~1kB each), doesn't seem too bad an option? (You'd need to use a class or data attribute to switch theme)