Is raceway wiring allowed in a bathroom?
There is nothing in the NEC that prohibits the installation of surface raceway (metal or nonmetallic) in even a damp or a wet location provided the conductors inside are rated for said damp or wet location.

That said, many AHJs do not consider the area outside of the shower zone to be a damp location to begin with.  So, go ahead with the raceway -- just remember to use a wire (such as THWN) suitable for a damp or wet location when running wires inside the raceway.