Based on advice from @jules and others, I finally got around to making a sticky mat for cutting fabric and paper. This is just a 12x12 inch piece of very flat 3/8" ply onto which I sprayed two coats of 505 adhesive…
Believe it or not, this piece was cut from a scrap of “grunge” cotton fabric. Weeding took far longer than the actual cut, and is one case where gorilla tape is contraindicated.