OK - so it should have dinged but didn’t. Twice.
I think the game got confused because you’re using a Monk instead of a Barb… yes, there is nothing in the achievement wording that says you have to be using the right hero class.
The thing is… you and I know that the Season will end on 18 Aug… do you want to take a chance that Blizzard will see this post and react fast enough for you enjoy your Galactic Wings ? I would start thinking of a « Plan B »… ie, level up another hero, and perform the last Tier 55 GRift with the “correct” hero class.
The IK set is one of the set I used for that achievement… with my barb… no issue at all. Perhaps you want to use that one, since you have all of the equipment.
Good luck !!