This is what I use in my car seat. It attaches to the seatbelt, not the car seat itself, rendering it much safer. It's called a PawRider and works with Chokefree harness or any other harness.
Ultra Easy ChokeFree & PawRider Pet Seatbelt Adapter (USA)
OOPs! I misread what you were looking for. My car seat straps in with the normal seatbelt for the car around it, then the pawRider is attached to the shoulder part. Will this work for your snoozer?