When you finished formatting and reinstalling, I would reinstall Origin and point it to the same location of the E drive. That way the set-up will recognize some files are already present and only add the relevant registry and system files to your OS. Then it should also be able to recognize the BF3 files, and if not you could download them and install to the same location, in which case Origin would only check the integrity and not have to download everything again. Personally I would delete all the files, as you can just download them later again without any problem... This way you can avoid any future issues caused by old files.
Whatever the outcome, as long as you still have your Origin username and password, your save game files should be safely stored on Origin's Cloud Service. If you are formatting your C drive though, I would backup the Battlefield 3 folder in your Documents, as that contains local settings for the game (like graphics and audio/key bindings) as well as local screenshots.