I'm not sure about Cozumel, but in Mazatlan and Puerto Vallarta the cabs don't have seatbelts, and they drive like NUTS!
We had our boys in Peurto Vallarta and just prayed everytime we were in a cab that we'd be okay.
So, I guess to answer your question...bringing the carseat would be a waste of time and effort. You will just have to be one of the Mom's that I scream at for holding their child in their lap in the car...it's really all you can do.
Have a great vacation!