I would also call SWA and add the lap baby to your reservation. We flew with our son at 3 months and 5 months. He goes again next week at 10 months. "They" say that they can only have so many lap babies on each flight, so once your reservations are made, it doesn't hurt to give them a call and add the baby so they know you have one.
For each flight, we checked in online 24 hours in advance like normal, but didn't print our boarding passes. Once at the airport, we checked in with him, got our boarding passes and showed his birth certificate. They have asked each and every time. He was born at 5.5 pounds, so at 3 months I think he weighed 8-9 pounds. There was NO WAY he was close to two, but they still asked for the BC. I had it ready.
Enjoy your flight!