This may sound silly, but I think it is important in this case: Have you asked your parents what they think? If not, you should do that immediately.
Also, since you can't find anything in the student handbook, I'd look for an opportunity to ask your principal what school policy is on this matter. Having a good understanding of the relevant Michigan laws would be an advantage, so long as you use your knowledge respectfully in the conversation.
As for what the Michigan laws are for students wanting to have a registered firearm in their vehicle at a private school, I'm not certain. You might want to post this question in the "Carry Issues" section of GT or wait for someone more qualified to reply here. But make sure you don't fall victim to the "I read it on the internet so it must be true" disease
Also, to avoid this whole dilemma entirely, you might seek permission to store your handgun at someone's house nearby. Maybe the father of one of your friends is an avid shooter and would be happy to keep your G22 safe during the day? To me, this would seem like a better option