Is the school's account showing and available for selection, but not actually connecting? Or is it not showing at all when logging into your personal account? Additionally, is the school's account setup as a group ora page (or both)? Per LinkedIn's API permissions, we can only connect with company-type pages that your personal account is connected to. Therefore it may be worth looking to this particular aspect of your school's account.
Otherwise, it may be worth reaching out to LI's support, to see if they can see anything on the school's account that would be preventing it from connecting with 3rd party apps - a permission setting or other account setting in that regard.