Single engine turbine
In your situation, single-engine turbine might not be a bad idea. Normally, I push multi PIC over single-engine turbine, but you have good multi time.
I have a friend who worked for Westair, whom I believe is a FedEx contractor in the Bakersfield area. He had very high total time and enough multi to get by, but no turbine to speak of. He worked there a year, built up 800 hours of Caravan time, and got hired by EJA. He said the place treated him relatively fairly and did not get surly with him if the weather or equipment weren't right. He also liked getting all the Southern California single-pilot IFR experience. Sorry, I don't have any contact info.
I haven't the foggiest if single-engine turbine PIC makes you less desirable for SWA. I believe that all they've ever cared about was the 1000 hours (and the 737 type, of course).
I think you may be on to something, bearing in mind, of course, that there are tons of furloughees and others who are several steps ahead of you in the quest for the Southwest job.
Good luck with your plans.