I am not sure on the laws regarding the "private" schools and whether or not the State made a "blanket" to cover all schools or just publicly funded ones.
I like to think of it like this...
Assuming you have a CPL or CCW
You can legally conceal/carry your pistol into a restaurant/bar if less than 50% of the establishment's income is derived from the sale of alcohol.
If they make 51% or more of thier $$$ via alcohol sales...it's totally illegal.
Do you really want to be that guy who is peeing himself while the ATF is doing a record of sales count if you were to be exposed or reported?
My advice is to play it safe and keep all weapons off school property. If you do happen to have it in your vehicle make sure it is as you described (in locked case, unloaded, and unaccessible to occupants in the vehicle...the trunk) Also tell noone you have it, as word of mouth kills!
It is an interesting question though...