After a lot of trial and error (more error than anything), I figured a way to cut a fairly thick stiffened felt. It's called "Nicole's bead backing" and it's a stiff-ish substrate for bead embroidery.
This technique has probably been referred to before, but I thought I'd also pipe in and say "IT WORKS" and you can use it on a regular mat! No need for the extra sticky one.
What I did is, cut a piece of label paper (the kind that has a sticky side and printable side) to the size of my felt piece and stuck that to the felt piece. I then brayered it all down on a regular mat, with the paper side making contact with the mat and the felt side on top. I did not tape down edges or cover the felt with anything. Voila, it cut like a charm. I did have to use quite a bit of force, extend the 60 plus blade a lot and cut at a slower speed but all in all I'm happy with the results.