I had the same problem with Renji for a while too. I uesd to keep him in his carrier, but he would always end up carsick, whining and generally disliked the car. I don't know what a snoozer seat is (though I imagine it is very similar to what I have), but I have what looks like a box with lining that has a strap that goes around the headrest of the seat and is up high enough for a small dog to look out the window. It has a strap that connects to his harness to keep him safe. Keep in mind, this is actually something I bought at the store, not a box I decorated, haha.
At first, he wasn't sure what to make of it and he would constantly brace himself on both sides. But now that he has gotten used to it, he LOVES the car and hasn't once gotten carsick or stressed out by being in the car.
If you just google search pet car seat, you will find tons of products that are exactly what I'm talking about. Good luck!