I see nothing wrong with that as a way to avoid girls being busted for having it in their purses. However, this might result in a logistics nightmare. Consider
Burbank High School in Burbank, CA. It has an enrollment of about 2,800 students. If we assume about half are female, i.e. 1,400, then if we can get a rough figure of how many of the them are menstruating at any given time we could see how many might be going to the nurse's office. According to
this website (and I'm not sure of its accuracy) about 25% of the female population is menstruating at any given time. Divide 1400 by 4 and you get 350. if we assume this 350 includes girls at all stages of their menstrual cycles and further assume that the girls only come in to see the nurse on the first day of their cycles, then we could dived 350 by 7 to get 50. So, every school day we may have up to 50 girls having to visit the school nurse.
Another problem I could see developing in such a program is that girls might fear being "outed," i.e. they might feel too self conscious about going to the nurse and having everyone know it's their time of month.
Of course, I'm speaking of this as an old male. If any of the women on this thread would care to weigh in on this subject, I, for one, would appreciate their insights.