If the wheelchair doesn't fold, prepare it as much as you can for being transported.
We have not used Southwest, but have traveled on a large number of different airlines and gate checked DD's wheelchair on all of them. We bring the wheelchair right to the door of the plane. DH usually is the assistant for DD to get her seated. I am the 'wheelchair wrangler and get it ready to go in the baggage compartment.
When we take DD's wheelchair, I remove
- the seat cushion (velcros on and could pull off and be lost),
- the seat back (the connectors could break if something fell on the back),
- the armrests (hers pull right off and could get lost if they get dislodged).
I can't remove the footrests, so I put them up and/or attach them together with the foot straps on them. I also fasten her seatbelt down and pull it tight so the seatbelt doesn't fall into the wheels and get tangled in the axels.
The last thing I do is fold down the back canes (the upright posts that the seatback is attached to). That makes it as compact as possible and removes as much stuff that could be damaged as possible.
I put all the small pieces into a laundry bag and carry them onto the plane. They go into the closet or overhead bin.