My 3 yr old yorkie Seri flys all the time with me on Southwest. I have a prescription from my doctor and carry her shot records also. I haven't had a problem with Southwest. They only carry SERVICE animals.
I flew Alaska and America West. They both cost me above $75 each way. I walked by Southwest and they told me if she was a Service animal I could continue to fly with them. Seri has been flying since she was a few months old. She's good at the airport. She walks through the metal detector like a trooper and waits for me to come through. Sometimes her ears hurt at the beginning of the flight but not always.
We were flying from N. Idaho to Las Vegas for a nasty 2 yr divorce. We also travel to So. Cal to visit the grandchildren.
Have fun with your baby flying.
