That isn't true. Public schools can get funding for students who are only enrolled part time. Actually, it has inspired some public schools to start up 'home school programs'.
You stated, Public schools can get funding for students who are only enrolled part time."
Participating in band does not qualify as "part-time" and isn't applicable in this case. It takes more than one Band class to qualify as a student in any sense of the word. The school would receive "zero" funding for a girl who just wants to take part in band.