You use it.
The key fits the Inspector's cottage in the Countryside.
You go to the inspector's house - the key unlocks the door.
You use the key from the Curator to unlock the Inspector's house in the CountrySide.
You now have a key that fits a lock on Counterfeit Island, and you haven't found a key to the lock on the Inspector's house, so...
There is no key to the chest, and it never opens. It is a distraction.
Once you get the key from the curator, go back to Counterfeit island and use it at the Inspector's house.
The key to the Inspector's house is given to you by the museum curator after you chase the thief on Counterfeit Island. The sign is rather obvious, since "Veuve Noire" is French for Black Widow (it is misspelled 'niore').
She warns you that there is a notorious thief at work on Counterfeit Island, and gives you the key to the Inspector's house in the Country Side.
You do not have to find the key. It is given to you later in the game.
Use the key she gives you to open the inspector's house in the Countryside.
If you have been to the Countryside, you could not get into the Inspector's house because you didn't have the key. Now...