So a friend of mine needed help clearing the last two waves of the robot arena and I said "I'll bring my Mechromancer" and whenever I would join his game, he got a "Disc unreadable" error. We double checked that he had both the Mechromancer DLC (He and I both pre-ordered the game) and he had the newest Compatibility pack, as I have the Season Pass and he does not.

I tried with another character, I got into his game fine, so as a last-ditch effort I switched Gaige back to her default head and skin (She was using the Scarlet DLC head and halloween skin). Default Gaige joined his game, no issues, arena cleared.

So, apparently, even with the Compatibility pack promo skins and DLC rewards can cause issues in multiplayer games.