Due Process Hypothetical

A public elementary school has an incident in which a child becomes very ill.  The child is allergic to peanuts.  The child ate a peanut butter sandwich in the school cafeteria because she traded the tuna fish sandwich in her lunch box for a peanut butter sandwich brought to school by a friend.  In the aftermath of the incident the school adopts the following rule:

No child can bring any food item containing peanuts or peanut butter to school as part of their lunch.

1) Is the rule constitutional under the Due Process Clause approach of Lochner?

2) Is the rule constitutional under the Due Process Clause approach of Nebbia, Carolene Products and Lee Optical?

3) Is the rule constitutional under the Due Process Clause approach of Griswold?