First off... why do you need to rename the movie?... are you referring to the .swf as "the movie" or are you calling the .flv file "the movie"?
Sometimes embed code uses Flash Vars where the .swf and the .flv are external and can be easily edited, other times the name and path from the .swf to the .flv are hard coded into the .swf... then you would need the .fla to edit the .swf... or just change the names of your .swf and .flv files to match the original code and file names you received.
Not really enough details for exact answer.
Also not sure what this means:
Do I simply rename the HTML page to the desired link name for brand consistency on my website, will that solve the problem without needing the FLA?
The HTML page and the .fla have nothing in common... not sure what you are getting at here.
If the movie is embedded in the SWF then you can't simply send a FlashVar to the SWF to load another movie.
Embedding a FLV inside the timeline of a flash movie is typically not a good idea. You should embed a FLVPlayback component on the stage. Then you can use the component inspector window to set the skin for the playback functionality. The source of the FLVPlayback should be passed into flash via the above mentioned method, FlashVars. Then you can set the name of the external video FLV the play.
If you have been given back a bunch of videos embedded in different SWF files then renaming the HTML page to something else is fine. Just make sure the embed code is pointing to the correct SWF to play with the correct embedded FLV. The names of the SWF or the HTML file can be changed at any time as long as the HTML file points to the correct SWF (and adjusts for any width/height differences).