EDIT: I may be a bit late to this party. But...
I was not aware that Southwest allowed birds in cabin. You may want to specifically call back to verify this to avoid being turned away at check-in. Most airlines that specify "dogs and cats" don't allow birds. Also, I would look into a hard-sided carrier if at all possible. I fly WestJet with my 'too, and they ONLY allow soft-sided mesh carriers which is a nightmare because the maximum lifespan of a carrier is one flight. I even travel with duct tape in my carry-on for emergency repairs and so far I haven't found a good carrier that will last me more than one flight. We also have to stay in a hotel for a night before we can cross the border to the US so I use a collapsible dog crate for him and it works really well.