There is a bit of a caveat to that.
From the SWA website, children 2-11 must be aged verified before they travel. They can pre age verify by sending in documentation to SWA so they can check in online, but at some point they have to show SWA a BC or they aren't flying SWA. This is for the Children or youth fares. I'm not sure if they would have to for an adult ticket, but given the security restrictions, it wouldn't suprise me.
http://www.southwest.com/travel_center/children.html
Just an FYI (and I know, off topic too)...We just flew SW in August and DS (age 2.5) did not need his birth certificate to travel since he was flying on an adult fare. DD (who was 9 weeks at travel) did need her birth certificate since she was flying as a lap child.
Also, the PP may already know this, but she mentioned 2 lap babies. Only one lap child is allowed per row due to the oxygen masks (I would assume with three adults that they could sit in different rows). If the 2 year old suddenly became a '23 month old lap child' then he/she too would need to be in his/her own row separate from the two other lap children.