So, I got the video to work.

The FV Pro Pack upgrade didn’t fix the problem, though it’s probably good to have both bits upgraded at the same time.

We decided to go HTTPS-only on this site, i.e. 301 all non-HTTPS traffic to HTTPS. As part of this I adjusted the siteurl to HTTPS. The video now plays as it did without security on it; i.e. the problem is resolved. (Thanks for your help anyway.)

So, basically, Flowplayer on HTTPS requires that your site have HTTPS as part of it’s primary Site URL; which matches with the direction the web is going anyway, so I can’t really complain about that. It might be helpful to note that in an FAQ or documentation somewhere for other developers (assuming Google doesn’t just pick up this forum thread).