After a LOT of thinking and research, I've finally come up with a solution that resolves my incompatibility issues with my retail Xbox 360s.

Let me explain this as best as I can...

On the client console, the game version was 44.0.-1, interestingly enough, whereas the host console's game version was 44.1833.17. To bring the client console to the same version as the host, I downloaded and signed into my Xbox Live profile, and let the game sync with the game servers, before exiting and allowing others to join the party with their own, non-Xbox Live, profiles. Doing this brought the game version to exactly the same as the host, 44.1833.17.

From my understanding, 44.0.-1 is a form of error code from Treyarch Servers, and thus rejects the game's attempt to sync with mainline servers due to the profiles signed in being non-Live profiles. As such, this can be overridden temporarily by using the aforementioned solution.