How does Billowing Smoke work with Greedy Dice or Jirachi {*} and its "Wish Upon a Star" Ability?
If you KO a Pokemon with Billowing Smoke attached, you do not get to use Greedy Dice nor Wish Upon a Star since neither of those cards would be going into your hand. Any Prize cards that would be taken for a Pokemon that has Billowing Smoke attached are discarded, not taken.
Source: Darkness Ablaze FAQ (2020-07-31), TPCi Rules Team (2020-08-13)

Relevant cards

Note: All applicable cards may not be listed below.
SM7_97
SWSH3_158
SWSH3_158