You might want to ask this over on the Transportation board! They're all very experienced DME people.
In short, I have used DME more times than I have fingers and toes. Using it again in 3 weeks...like you, flying SW with 6 checked bags!! I'll be tagging those bags and saying goodby to them at my home airport.
Could your bags be misdirected by SW? Sure. But probably not. Are you flying direct...no changing planes? If so, I really wouldn't worry about it!
But...if you're going to stress, it might be better to get your bags yourself. Believe me...plenty of people bring huge amounts of checked luggage to the bus with them! Of course, I've also watched some of those people attaching the yellow luggage tags to their bags, as they are in the bus line.