If you are eligible to use Magical Express, it doesn't matter which airline you use. ME is an arrangement between you and WDW, not your airline. When you register for your Disney resort, they will take your flight info. When you arrive at Orlando airport, they will already be waiting for you and your bags as they know your coming. SWA, or any other airline has no bearing on this service. WDW will retreive your bags from any airlines baggage claim and deliver them to your room.
On departure they will drive you and your bags to the airport. From there you will have to check your bags with your respective airline at the ticket counter. If you wish to use ME and are staying at a WDW resort, then make sure res has your flight info. If they dont know when your coming in, they cant be there to pick you up!