Sure, carry-on the car seat base. I've done that many times.
Just confirm that the total HxWxL of the box meets the airline's dimension requirements for your free checked bag. I know one of our carseat/base contraptions came in a huge box that might have been considered oversize by airlines.
We have one or two family members who occasionally assume we can take things for them on flights. They think we're EXP or 1K or whatever and can check a virtually unlimited amount of stuff. Star Gold and AA Gold don't go nearly as far as they used to...
