I had Black Ops 1 and the Moon map pack for Xbox 360 and I recently downloaded it onto my Xbox One. However, my DLC did not download. I checked the store to see if it would let me just download it but it makes me pay. Is it not possible to download the DLC I already paid for?
Similar to the My Games & Apps menu that shows you all content installed on your XB1, you can see what content you have installed (and ready to install) on a per-game basis.
To do this, press Start on your backwards-compatible game's tile (on the Home screen or in your game library), and click Manage.
This should show a screen similar to your games library, but the "Installed" and "Ready to Install" should just be for your game. Any DLC you own should be in the Ready To Install category, as in the screenshot below.