Be sure to keep the ac on at all times, even when you stop to eat or go to the bathroom. Keep the cage in the middle of the back seat where it will be less likely to ever be in the sun. Position the ac vent to blow right at the cage. Pack a cooler with some frozen water bottles and any cookie tins or chin chiller (marble tiles) you might have. You never know when you might have car troubles and if the ac went out or the car died these cold items rotated in the cage could save your chins life. Always better to be safe than sorry.
Keep the radio in the up front speakers or turned down low if your chinchilla isn't used to that much noise. If you keep him in his cage be sure all large tubes and houses are on the bottom floor or are anchored down.
If you choose to use the cat carrier change the shavings in his cage about 48 hours before the trip. Then the night before the trip fill the cat carrier with the shavings from his cage. Since they will smell like him he will be more comfortable. Place the chinchilla in the cat carrier the night before also so he can settle in while he is still in familiar surrounding.
He will be just fine