My daughter plays violin/viola but she travels constantly with cellists. They always book a seat for the cello. As far as I’m aware Southwest allows cellists to buy the instrument a seat, but I’d advise both calling them (which you are trying to do right now), emailing them, and also doing web research on problems with Southwest. Even airlines that are usually “safe” for instruments sometimes make egregious mistakes when a flight crew is uninformed about their own policy. While it is possible to check a cello if you have a flight case, as i’m sure you already know, doing so presents a big logistical problem at the other end because you have an enormous case to deal with until the return flight. As an aside, bassists in her traveling groups usually arrange to borrow an instrument at the other end.
[edit]: A lot of instrumentalists I know keep copies of the airline’s flight policy printed out in their cases. My daughter for one keeps a certificate from our local international airport that confirms her ownership of her instruments/bows–this is mainly for international travel, though.