I just went through all this myself, and here is what I found out. . . Tell origin to keep the downloads, make a copy of them! The downloads are in a folder HERE: C:/program files/origin games/ and then all the EPs and SPs you have downloaded are there. I copied and saved these folders to an external hard drive and when I need to reinstall, or move the game to another computer or whatever, I download origin (After using a program called REVO Uninstaller to completely remove everything so I have a clean installation) and put my game folders that I have on the external back in the program files/origin games folder. From there Origin says it is ready to installl them. And in a few minutes, you will be up and playing again.
One warning, you will need the manual patches to reinstall your game. 1.22 comes after base game and 1.36 goes after Late Night. Without those you won't be able to continue installing.
Also, once in a while, moving a file will corrupt part of it, you could end up removing and redownloading one of the parts of the game. Still, redownloading one expansion or stuff pack sure beats having to redownload everything!