I've turned green canvas micarta on a wood lathe before, to make 1/4" pins out of scrap. It actually went well. I had to go slow and use a pretty light cut, but I did use Craftsmen lathe chisels (sharpened a couple times during this job), and after a light finish pass with the skew, got a decent finish. I maybe could have gone at it harder, but the little lathe I was using would power out or slip, and the chisels probably wouldn't have liked it anyway. Your wood lathe looks a little stouter than mine.