When I try to load the game, along with the Artorias of the Abyss DLC, to play offline, I receive an error stating the DLC failed to load properly and that I will not have access to that content or to my characters who have DLC equipment. However, when I load the game and DLC to play online, it works perfectly. Can this DLC only be played online, or is something wrong?
Whatever Xbox you downloaded the content to first is the Xbox that it is licensed to and should be able to be used online or offline from anyone who uses the Xbox locally. If you have downloaded the content to a different Xbox first, this is why. To fix this, try deleting the content and redownloading it. If that doesn't work, you will have to re-license the content from the Xbox or the link on Xbox.com
Microsoft requires anyone using a Xbox360 to have internet in order to access DLC for a game whether it has online features or not.
If you do not have access to an internet connection, the DLC will become unavailable for use and all save files that have utilized the DLC will become unplayable until you have access to the internet again.
Unfortunately this is the system Microsoft implemented to prevent people from sharing DLC across multiple consoles, but it affects the owner in the same way, no internet, no DLC.