In Washington you can get a trip permit that allows you to drive a drive an unlicensed or expired car for a few days to get your business done. The cost is only $6 or $12 or something like that. I imagine CA has a similar thing.
To be fair though, why would you be allowed to drive a car with expired license, renewal paid for or not? You were notified that your car needed an emissions test, you had plenty of notice that your license was going to expire, why wait until the last minute and then ask for an exemption. Why not go through emissions 2 weeks before the license expired to make sure you had enough time?
Perhaps my attitude is tainted by being on the other side of this experience. I often get someone coming in at 3:00 on a Friday afternoon with an “emergency.” It seems the car license expires today and they have to get through emissions. When I tell them I’m full for the rest of the day they often huff out complaining that they’ll go find someone who wants to help them.