Bone constraints for shaving sequence

I’m animating a sequence showing a character shaving. I want to constrain the tip of a hand or razor bone to follow a path along the surface of a cheek or jaw, and but also constrain that bone’s rotation to another bone, to keep the hand oriented straight up throughout the sequence. the Copy Rotation constraint takes care of the rotation requirement here. But, since the hand bone is connected to other bones, it won’t obey a Clamp To constraint telling it to move along a path. How to I make the hand follow the path?