Take your profile and game data on a USB drive with you.
Recover your profile on the other 360, and re-download the Online Pass. This should be available to you on the other machine since it was previously downloaded. The full steps to re-download content are here, but basically: Guide, Settings, Account Management, Download History, find the item and re-download it.
This all being said, I dont' have the game and haven't tried it, but let me know if it works.
Best Answer
Yes, you can use EA's Origin store/application to purchase and download Battlefield 3 here.