Don't want to drive your own car? Rent one from an outside vendor.
Current cars currently include a 2006 Mazda MX5 (NC), 2010 Mazda RX8, and 2013 Hyundai Genesis Coupe Turbo with paddle shifters, perfect if you prefer no clutch pedal.
Cost is a very reasonable $800 for a full day or $500 for half day. The car will be waiting for you at the track so all you do is jump in and drive.