They won't care. I've emphasized that in my response. I'm not disagreeing with you here. Without extra info on the additional closet that is being added, I do think it's not an issue.
One of the main reason(s) that the state wants you to apply for permits on additions is it raises the value of the property. That means more property tax the state can receive in the future. With this closet, the addition to the home is probably near 0. Which is why I would argue it's almost a waste of time for the inspector to come as it generates no revenue for the state. The cost for the inspector to come outweighs the return to the state in this case.