I'm cataloging my console's physical and digital purchases on LaunchBox as well and I like to keep track of the DLCs I own. A proper feature would be nice indeed but here's my way:
When I first started adding games to Launchbox, I simply listed all the DLCs in the Notes field, just after the game's Owerview. This is a very quick solution and the Notes can hold a lot of text; however it could still turn a little messy when there is a very long list of DLCs for a single game. I had also to be careful when changing or updating the game's name or ID metadatas since this could revert the customized Notes field into the original from the LB database and thus deleting the DLCs listed there.
Lately I'm using a different approach: for each game I create a text file with the DLCs list and then I add it as an additional app (I'm storing these files into a DLC subfolder inside the LaunchBox root). With much more freedom to customize the list I'm now grouping together the DLCs from a specific Pack or Season Pass, adding color text, specifying if the DLC is already included on the game disc, if it's a free or paid content and if it's compatible with other versions of the game.
I also created a "DLCs" custom field in Launchbox with a "Yes" or "No" that will appears in the details panel. This way I can check if I have any DLCs for a selected game and I can also search how many of my games have DLCs in the left Side panel.