Can I use *free* DLC while Mass Effect 3 is offline?

Well as with past experiences with DLC being downloaded from a different xbox but with your home hard drive, I would say this probably will not work. If you put your hard drive back into your home console and try to play offline the DLC will be "corrupt". All this means is that the DLC is only available for offline use on the console it was downloaded from. I've had this problem with multiple DLC's before i knew the cause. It has something to do with network keys or something of the like. Even now if you were to delete the DLC and try to redownload it on the same GTAG, it will still be corrupt. One way to remedy this is create a new Xbox live profile on your console and hard drive. Then download your DLC with that one by following the advice below.

My advice is next time you want DLC just send your xbox over for the day or go over for the day.