OK. I’m just comparing to my BL2 game right now. Make sure you’re signed in as the account used to purchase the content, then start the game and go: Main game menu > Downloadable content.
When I select Headhunter 3: Mercenary Day, the entry says both “Purchased” (top-left of the tiem “card” that pops up when you scroll to the item) and “Installed” (top-right). While you’re at it, just double-check that you have only ONE compatibility pack installed (it may be easier to check that through Console > System > Storage > hard drive > Ganes & Apps > Borderlands 2)
If yours doesn’t match, try re-downloading it then fully quitting the game and restarting. If it still fails to say “Installed”, you may need to clear your system cache and try again.
The one time I had major issues with this was after replacing the 360 and migrating everything to the new one; I ended up having to delete all the content, backup the saves, clear the cache, do the licence transfer, then re-download everything. Hopefully, you can get this working without going to such lengths!