Surely his instructor is a CFI
Fair enough, over here that would be the Chief Flying Instructor i.e. the boss who would be able to reallocate the student to a different instructor, or even take him on himself if need be.
I'm not familiar with the FAA system or stage checks, but if the student is at a civilian flying school he is the customer, and if he is not happy with what is being provided he is well within his rights to question it.