I know I need to have BD/SP save data, and I've finished BD. Do I have to beat Legends: Arceus as well, and if so, how do I access the quest?
Darkrai may be captured near the Clamberclaw Cliffs with any amount of save data from BDSP. To encounter it, you must first finish Mission 18 -- the one that has you capture Dialga or Palkia at the Temple of Sinnoh. After this, Mission 93 is the one that starts the Darkrai quest.
Darkrai (Bulbapedia) Clamberclaw Cliffs (Bulbapedia) Task (Bulbapedia)