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I would suggest asking the Superintendent. If he is given specific permission, then he falls under the "pursuant to written regulations or written authorization of the institution". It could be given on a temporary basis and limited to permission to have the weapon in that specific school vehicle for a specific time frame. I would certainly want everything in writing.

Our school district gives authorization to certain staff to carry at school, school functions and in school vehicles. Others are permitted to carry at school but not at athletic events, etc.