G-CPTN,
Flash is trying to use temporary storage, not to permenantly grab space.
Next time you go to the BBC web site, or YouTube or any other site serving adobe Flash content, put your mouse over the video portion of the screen & right-click your mouse.
You should get a menu telling you about Adobe Flash player & a "Settings..." option (the top line of the menu generally tells you what is sending the video, on the BBC site it tells you which version of the BBC media streamer is running - don't worry about this)
If you select settings you get a small popup with tabs at the bottom that you click to see what options you have. In the middle is a folder symbol and in there is a slider that allows you to allocate temporary space to Flash. If it is set low, then you may want to increase it - on this computer I've allocated 100k which is the same as you have. For some reason BBC streamer wants more, it may be a HD format video. You can set the amount used there, 1MB is neither here nor there, so I'd suggest upping the limit and then the message will stop