I finally found a way to do this with FFmpeg. The trick was to disable audio, use a different flv meta data analyser and to reduce the duration that FFmpeg waits for before processing. My FFmpeg command now starts like this:
ffmpeg -an -flv_metadata 1 -analyzeduration 1 ...
This starts producing frames within a second of receiving an input from a pipe so writes the streamed frames pretty close to real-time.