We had the same problem with drop off a few years back with National. Our 10:00 PM departure was delayed until 5:00 AM. after several phone calls to the station manager where we were told the after hours drop off was our problem. After stopping by the location at around 4:00 AM and finding no one there. I took the car to the airport parking lot and parked it as close to the exit booth with the keys and parking slip in the arm rest.
After the location opened I called them and told them where the car and the keys were which they seemed to be fine with. A few hours late I received a rental receipt via e-mail with only the cost of the extra hours and the parking cost added to my invoice.