I think the school knowing that you carry is best for everyone. Suppose it is stolen? And I don’t consider giving the school notice of your carry is an infringement.
I don't think "giving" your school notice is an infringement either, but that's not what would happen.
You would be forced under penalty of law to give such information. It won't be voluntary. You would be threatened with fines and/or imprisonment, even for inadvertent violation.
It is absolutely NOBODY’S business if a person chooses to carry. To carry or not to carry is a personal choice, and (save for the no felon prohibition) is not subject to governmental oversight regulation or rule, IMHO.
What if it is stolen? Any competent gun owner would have the serial number written down, and should report it stolen. End of story.
With respect to giving the school your “notice of intent to carry” is an inFRingement, “Excuse me?”
The beauty of concealed carry is that no one knows you are carrying. That is a reason, in and of itself, to not advertise your carry. To anyone!