How terrible!
One of my cats did that to me on the way TO the vet, then again on the way home FROM the vet. At the time I had small children and had the car seat in my old heavy chevy truck. I had no pet carrier so I just put my kitty in the truck and off we went. The first 9 minutes were great. At first glance I though it was cute that lil girl (cat's name) was sitting in the car seat. How sweet I thought.
At second glance I noticed that undeniable bow in her back and ears pinned back in $hit mode. Then just like you said, the smell hit. This happens at the very second I make a left hand turn into the vet's parking lot after a 10 minute drive.
On the way home, same thing (but not in car seat this time) right as I'm pulling in the driveway she has diareah on the seat so I knock her into the floorboard where she has more diareah, then pukes everwhere to top it all off.
Man that cat has some serious trouble keeping her bowls in order whenever she rides.
I know your pain. I cleaned all of this myself. Luckily no carpet and the carseat cover could be easily cleaned. You should be ok for at least 5 more years before something like this can happen again.