The quest is completed by returning The Rueful Axe to the Shrine of Clavicus Vile and not killing Barbas. Killing Barbas will reward The Rueful Axe instead (note that receiving the axe will NOT count towards the Oblivion Walker achievement).
The display can take both the original and the replica.
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If you returned the axe to the shrine, you only need to have had the masque in your inventory for the replica recipe to be available. If you instead chose to kill Barbas and not return the axe, the replica recipe will not be available.