Is a 6 foot 5 inch ceiling height ok for a basement bathroom?
The Code ( ICC R305.1 ) states that habitable rooms shall be 7’ high and bathrooms shall be a minimum of 6’-8”. 

However, there are 5 Exceptions:

Exceptions: 


For rooms with sloped ceilings, the required floor area of the room shall have a ceiling height of not less than 5 feet (1524 mm) and not less than 50 percent of the required floor area shall have a ceiling height of not less than 7 feet (2134 mm). 
Not more than 75 percent of the floor area of a bathroom or toilet room is permitted to have a sloped ceiling less than 7 feet (2134 mm) in height, provided an area of 21 inches by 24 inches (534 mm by 610 mm) in front of toilets and lavatories has a minimum of 6 feet, 4 inches (1931 mm) in height, measured from the finished floor. An area of 24 inches by 30 inches (610 mm by 762 mm) in front of and inside a tub or shower shall have a minimum of 6 feet, 4 inches (1931 mm) in height, measured from the standing surface of the fixture. 
Beams, girders, ducts or other obstructions in basements containing habitable space shall be permitted to project to within 6 feet 4 inches (1931 mm) of the finished floor. 
Beams and girders spaced not less than 4 feet (1219 mm) on center may project not more than 6 inches (152 mm) below the required ceiling height. 
Conversion of existing nonhabitable spaces, such as a basement or attic, to habitable space, shall provide a minimum 6 feet, 8 inches (2032 mm) ceiling height for flat ceilings or the portion required under Exception 1 of this section.