Is my bathroom project prohibited due to a low ceiling height?
It may be possible to be granted an exception regardless of what code states depending on the flexibility of the Authority Having Jurisdiction.

Ask the Authority Having Jurisdiction. In other words, ask your local building department or whomever has authority to approve permits where you live to see what they require and whether they would permit such an exception. 

For example: When I was finishing my basement, I have a point that goes below minimum required ceiling height for my state's building code. My local building department follows my state's building code. My state's building code is generally based on the International Residential Code, but has a few exceptions (one noted difference in my state is the required minimum height for finished ceiling heights and protrusions when finishing pre-existing unfinished basements). My building department granted me the exception I listed in the submitted plans. This example also shows exactly how locale-specific this question is.