I got an expansion pack via digital download and now that I'm installing my Sims games on a different computer, I want to get that game on it as well.  The thing is, the site has no idea that I've downloaded that game even after logging on, and it looks they've switched from EA download manger to Origin.  Does anyone know anyway to get the game without paying another $20/40?

Email them.  I've never had the issue, you can the same code you got with the download to register it on another pc (you just won't get the game bonuses that came with it but they're linked to your account when you first redeemed them anyway) but if the game isn't actually appearing on the site to allow you to download you need to inform them and tell them to sort that shit out.