If we were to edit a texture for a static character will it always require to have the LRC cleared to be able to see the new texture?
Possibly not. Priming the LRC with that asset might do the trick, though I am speculating. I don't know if you are familiar with doing that, but fyi priming is to add to the LRC via the Repo Browser and would be done so that you can get the asset into your LRC when it is convenient instead of when you first encounter that asset in the game, which would otherwise require downloading it then.
Anyways, the repo browser has a a menu Configure-->Settings, then check Prime Local Cache. With that checked, when you drag an asset from the repo to your local folder, it will look like you are copying the file to that location, but it will actually be updating the LRC with that asset. If you look at the LRC location*, you can actually see it being updated.
I have only done that for adding things that were not already in; I have no idea if it will update when the asset is already in but the repo version is newer. It is worth a shot, though.
*For me, on Windows 7, it is
C:\ProgramData\FOREVERINTERACTIVELLC\Repository