How can I address a student systematically boosting their grades by filing frivolous complaints?
When it comes to "he-said, she-said" situations, as seems to be the case here, most institutions will tend to take the side of professors by default. For obvious reasons—the 18–22 year old college student is much more likely to lie to obtain a favourable mark, than the professor is to act maliciously to sabotage a student. The only scenario where the professor might be in trouble is if there are accusations from multiple students. 

The most significant red flag here is that the student is making essentially the same accusation against nine different professors. It is preposterous that nine professors would collectively decide to sabotage a single student. 


  So far I have read the students' code of conduct carefully. There appear to be nothing that can be directly applied here. The closest appears to be the rule about dishonesty, but it wouldn't work by itself.


Making dishonest accusations to obtain more favourable grades is an unequivocal form of academic dishonesty. Any respectable institution would discipline a student engaging in such behaviour. If this behaviour cannot be disciplined at your university, I would suggest seeking employment elsewhere. 

You should lodge a formal complaint, possibly in collaboration with the other professors who are being accused. You have a rock-solid case.