Discussed this issue with Laurent directly through support channels. The main problem here was that websocket (used by the flying-sphinx gem) uses autoload, which doesn't play nicely with Sidekiq. Switching the production Rails environment to be threadsafe must tweak something in autoload, because it fixed this problem.
So: if you're using Sidekiq, make sure your environment has threadsafe enabled, and even better, don't use autoload (Matz is planning to remove it after Ruby 2.0).