Found a way out! The secret "double tap" gesture described
here got me back to the Kindle menu, from which I could perform a factory reset and get back to regular user mode.
I'm still unable to register the Kindle. Apparently, a Kindle OS downgrade might fix this issue, but it's unclear to me if there is any viable downgrade path for a Kindle running 5.14.3.0.1. Does anyone have a reliable solution?