As stated in the title the Quest mechanic will not let me progress any of the Grail Knight branches despite buying and using the Grail Knight a good bit now. I would just like to be able to unlock some of the better cosmetics for using the class.
What @Heretical_Cactus said. There are 2 tiers of the Grail Knight DLC. A very cheap base version that unlocks the class and all its gameplay elements, and a version for a little more that also unlocks the cosmetic components of the class, like the challenges for frames and skins.
So, if you want the cosmetics, you need to upgrade. Good news is that you don’t need to redo the challenges you already did if you do, you’ll be able to claim it as soon as you bought the upgrade. (At least, it works that way with the other DLCs, so I expect it to work like that with this one as well.)