What card is the inspector trying to play? Is the issue having overcurrent devices in a bathroom? If so, I don?t think that rule is very old. It goes back at least to 1996, based on the books on my shelf. But if the house is older than that rule, then you have an existing condition that was allowable at the time of construction, and the inspector cannot force you to change it now.
As to whether this is a bathroom, my answer is no. The NEC defines bathroom using the word ?basin,? but it does not define ?basin.? My desk dictionary defines ?basin? as ?A SHALLOW (my emphasis), usually round container or pan, used chiefly to hold water or other liquid, especially for washing.? The common bathroom sinks would meet that definition. But a laundry tub is not shallow, and it is not chiefly used for holding water. Nor is it intended for washing, at least in most homes. A laundry tub is chiefly used to as the place to dump water from the washing machine, and the washing takes place in the machine (not the tub).
Thus, IMHO, since the laundry tub is not a basin, this room is not a bathroom.