Yes, you can ship it on Southwest. That's what I would do because it just counts as one of the two bags you can check for free. I've done it a couple of times.
If shipping on Southwest I would use real ice and not dry ice. If you use real ice, it only has to be in plastic bag inside the cooler. If it's dry ice, it has to be a vented container.
I put the ice in plastic bags and hardened it up in a shape that was as close to the shape of the cooler. Put ice bags in cooler, ribs in plastic bags (trash bags) and put them on top of the ice. Cover with towels or whatever to take up any air space, Wrap a strap around the cooler and check it as luggage.
I'm assuming the ribs are still cryo'd so putting them in a trash bag will be okay. As long as the cooler is less than 50lbs you'll have no excess baggage. Either way,it's probably cheaper than buying dry ice and shipping UPS or Fed Ex.
Hope this helps.