Check out before you go to your interview early in the morning. Tell the front desk that you want to leave your luggage for pick up when you come from the interview.
For example, you have to be in the department for the interview around 8:30 AM. Check out around 7:30 AM and leave the luggage in the hotel. Pick up a taxi and reach the hospital 30 min in advance. However, do not show up immediately but have a cup of coffee and then go to the department at time. Make sure you keep a "safe margin" of time between leaving the hotel and reaching the hospital. Consider the rush hour in the morning, delay in the taxi arrival,..etc. I would rather be in the hospital early than be there late.
You are done at 4 pm. Pick up a taxi back to the hotel. Take your luggage and head to the airport.
Hotels have a way to keep the luggage for customers. You will be paying the hotel fo one night only. I did many interviews and I didn't rent a car. I arrive one night before and leave the following day. I use taxis between the hotel, hospital and airport.