Southwest Airlines is on side A of the Orlando airport, and Magical Express is on side B. So you'd have to go downstairs to baggage claim to pick up your bags, then go back upstairs to the tram, take the tram with your bags to side B, then go down to ME on the other side of the airport. Since you'll have a washer/dryer right in your room, I agree that it's worthwhile to pack light & occasionally wash clothes during your downtime. It will spare you the hassle of checking in the bags both ways, as well.
R.I.P. Disney's luggage services. So many of us loved you.