When I flew with my daughter in an infant seat, I took the base in a checked bag padded with clothing. At the time, I wasn't comfortable enough with the baseless install to imagine doing it several times a day. However, now that I have more practice, I find it almost as easy as clicking in the seat to the base and I would definitely never go through the hassle of bringing the base again.
I would read your manual carefully and practice the baseless install at home. Then try using only the baseless install for a day. If you can go baseless for a day at home and it's not too difficult, leave that base at home! If you are just not comfortable without the base, then I would pack it carefully in a suitcase padded with clothing. The nice thing about southwest is that I think your checked bags are free? That's a huge bonus when trying to bring extra stuff. Don't forget that if you have bought a seat for her, you get all the benefits (checked bags, carry-ons) of that seat as well as your own.
Oh, and the install on the plane is crazy-easy. Feed the airplane belt through the baseless install belt guides, buckle it, and pull the tail tight. Give it a good shake at the belt path to make sure there isn't any movement of the seat, and you're good to go.
Have a great trip! Flying with an infant in their own bucket seat is the easiest travel you'll ever get to do with kids, so enjoy it! My favorite part was that I could put my purse and diaper bag under the seat in front of her car seat and stretch out my own legs!
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