First, you need to sit down and read the policies word-for-word. Don't rely on phone calls or off-hand remarks from the credit card issuer (bank) or from your University.
The biggest hole in your coverage is likely to be personal liability in the event that you injure another person or cause substantial damage to some third-party's property. You may need to purchase supplemental coverage from the rental company for a one-off such as this.
Unfortunately, the information you have is not sufficient for a definitive answer. Remember, that if something does happen, it is the specifics which will matter the most.