I broke the corner off my 6 x 24-inch ruler. I saved it because I use those handles on my rulers. If I wanted to roll up fabric on my ruler, I had to take the handle off. Now I have one designated for that purpose. The broken corner isn't a problem.
12-1/2 square is a hard size to think about what to do. Maybe wrap it in lots of batting and fabric and make a little ironing board out of it to use next to your machine. Just make sure you use lots of batting so it doesn't melt!
Or . . . just throw it away. I'm like you though, I hate the thought of adding more to the landfill, especially plastic.