Student accused of Plagarism
Rather than arguing it with the prof, who seems disinclined to do so, let it go to the review board and argue it there. The student doesn't actually need to do the analysis them self before they have a review scheduled, but then, you can both insist that you have full information in which to defend yourself. Serve as the student's advocate if you are so inclined. 

If the student uses well designed code as they were taught to do, there may be, in essence, only one way to properly express the solution if it is small and well defined. That won't be true for a large program, of course. 

But also insist that the student should get the benefit of any doubt as the overall impact on the system is small, but that on the student is large. 

I agree with the advice you have so far given in this, by the way.