I also had the 737 type (and was in the SW pool) when I interviewed at FedEx. It did not come up in any portion of the interview, however I was prepared to discuss both the type and the pool if they came up.
I also second the benefits of getting the type even if SW doesn't work out. For me it was my first time going through a civilian course of instruction and simulator checkride. My only other experience with the FAA was during my ATP ride in the aircraft.
I also had never flown a Boeing product and it was helpful to learn their method of doing things, which made going through the 727 FE ground school at FedEx that much easier. Simple things like the switches on the overhead turning on in the opposite direction from what I was used to took quite a bit of time to re-learn and accept (I'm still working on that by the way).
I also felt a great deal of personal satisfaction knowing that I had completed the course on my own, a good confidence boost during the trying to get a job phase of my life.
Not having the type before I interviewed probably kept me from working at SW. I had too much time invested at FedEx to jump ship when the call for a class date finally came. I would have saved myself at least a year in the pool (estimate) had I gotten the type before I interviewed.
I would strongly recommend getting the type if you have a serious interest (as either your primary or secondary career goal) in working for SW if you can do so without hurting your family financially. You'll feel more confident during the interview and you will save a lot of numbers on the seniority list or months in the pool after you get the conditional offer.
Best of luck.
FJ