Don't know much about Firefox,but if you use Opera,you can get the Web Developer add-on.
It gives you just about every option you can think of accessing web pages.
Opera Browser Wiki :: Web Developer Toolbar & Menu for Opera
Sometimes with javascript,the files are broken into segments,and near the end of the segment,there is a referrer to another link.
This is to prevent you from recording the stream.
So even if you find the location of the beginning of the file,it does no good because it is only partial.
While the first segment plays out of your cache,the new link loads the next segment and starts at the same time the first is deleted.
If you find an app to capture these segments,and capable of putting then all back together,you might be able to save the stream.
Using linux,all flash streams,things like youtube,etc., are put in /tmp.
Not sure where they go in windows.
Here,they do not contain an extension in the filename,but you can still save the file,then give it the extension .flv.
Then the file plays just fine with whatever app capable of playing a flash file.
Some hidden files are put under /home/user/.macromedia/,the same technique works for these files.
If you find where the streams are sent to cache in windows,be prepared to copy or move the temp file before the stream ends in your player.
Some sites delete the file as soon as the stream has finished playing to prevent you from saving the file(s).
So if you go and look for it afterwards,where you know it should be,it's not there.
But it was.You just have to be quick.
I don't know anything about the windows apps for recording streams,but I know there are some out there.
I've seen them when I used to use windows.
It seems though if you are paying for this stream,the site shouldn't be all to worried about you stealing it anymore,and javascripting all the links probally wouldn't be enabled.,and these apps should work.