NPC doesn't recognise an achievement and cannot purchase an item as a result

[Canadane]

IIRC, its not actually owning the achievement that gives you the expanded inventory, but having completed the FATE successfully, where it remains like that for everyone before the FATE chain restarts. Once the chain restarts, you lose the inventory.

This is correct.
It's part of the FATE events, unrelated to the act of completing it itself.