No, you can't use Drunkard's Luck to gain advantage this way
My take is:
Since the DM calls the rolls, the negating the advantages and disadvantages happens in the DM's head*. The wording of the feature "when you make X with disadvantage" suggests that this comes at the end of the DM's decision. It also make sense for this to happen at the very end right before the roll because you are spending your precious resource and should have all the information related to the roll.
So how I'm saying this works is:
DM takes all the circumstances into consideration and calls for an X roll -here he says with disadvantage or not-. If DM calls for a roll with disadvantage you can choose to spend your ki points to cancel disadvantage.
*: The advantages and disadvantages all happen at the same time, then the DM calls it and players change the outcome with spells/features when possible. There isn't really an order of getting advantages or disadvantages. They are all either there or not.
But specific beats general?
Specific over general is if rules conflict, these rules don't conflict.
General: you have adv and disadv, they cancel each other out.
Specific: if you have disadv... you don't, fine.
Specific beats general is for when rules overlap, or conflict as i already said, it doesn't work the same for order of things. General happens first, specific alters what it can.