Replying to both responses at once - thank you both for the advice. I've actually stumbled on the solution with a machine I bought at an estate sale over the weekend. It's another vintage Singer, which when cleaned up and oiled - with a new heavy duty needle, sewed the Neoprene perfectly. I did a wide zig-zag stitch, at the very edge, kind of over the edge, and it gave a beautiful seam.
I will use the tear-away stabilizer to help with a clean start. Thanks again.
It may be the needle, it may be the thread, it may be the machine - but I'm not going to question the good fortune. The irony is that I didn't intend to keep the machine. I was buying the case/desk it came in. Beautiful, and I wanted it for my other vintage. Instead I now need to keep it because it worked on the Koozies. Go figure....