I have never checked the car seat, but have always carried it on for my daughter to sit in. Of course, we have always bought a ticket for her, but according to southwest.com:

The following items are not considered carryon bags or personal-type items and are not counted against the “one-bag plus one personal-type item” limit:
- A child restraint device for a child who has been ticketed, who has a seat reserved, or for which complimentary, available space exists.

So, even without a ticket for your child, there is a good chance that there will be a seat available unless the plane is completely full. Just realize that the car seat will have to be put in the window. Otherwise, I would attempt to gate check it just as you would do with the stroller.