Thanks everybody for the input. I've learned a lot about this stuff and how to cut it and how not to cut it!
Not having a plywood blade on the table saw, and having a seriously out of date jig saw (50+ years), I got to thinking. What blades do I have with my oscillating tools? I remembered that I had a semi-circular one that I assumed was for drywall (which I always cut by hand). That blade has lots of tiny teeth and has printed on it that it is for thin wood and plastics. Ah ha!
I gave it a try, using all the precautions I learned from the above posts, and voila! It cut great, no melting (used on about medium speed, using several passes on thin cuts), and the corners were smoothed with a bench grinder's fine stone on slow speed. Looks great.
It's certainly not a method that I would use for production speed cuts, but to cut out a 4" x 6" piece, it worked just fine. I hope my experience will be useful to those reading this thread, and again, my thanks for all of your input.