Perpignan is a very small airport, with just a few flights. (For example, today there were only 4 departures and 4 arrivals). That means that the car rental desks are only open for very limited hours.
National Citer's desk is open between 0930 and 1015, and then between 1400 and 1445, daily. So you should be able to pick up a car if you book the time as, say, 1430 (rather than 1500). If you can get back to the airport rather earlier (or later) you'll also be able to drop it off:
http://www.citer.fr/
It's likely that, by slightly tweaking your booking times, you'll also be able to book with other firms.
Chris