While the car seat bag might keep the seat cleaner, it will not keep it from being damaged structurally. After watching a luggage truck drive over a car seat, stop, toss it back on the truck, and drive off, I can't recommend checking a car seat under any circumstances.
Also, do not rent a car seat from a rental place. You don't know the history of the seat so you don't know if it has been crashed, you have no guarantee they will even have a seat available for you (I have heard many stories of people who had a reservation, but no seat was there), the seat may be completely inappropriate for your child's age/weight/height, and it also may be filthy/expired/damaged.
If you need a seat on your trip, plan to use it on the plane or purchase something like the Safeguard Go that you can easily take with you on the plane.