Have you ever owned 2 Xboxes? Then you probably need to do a License Transfer so you can use your content offline. Normally you can only use content you’ve bought on another console if you’re online with that gamertag. If you’re on a different gamertag or you’re offline, it won’t let you use it.

This. Plus, have twe identical USB sticks: TU#, save game, cache and gamer account. Leave only DLC on the HDD: smooth sailing from that point on (though you still need to do redownload those licenses, but it’s not because of customs, it’s because you’re offline).