Unfortunately the Stika is not meant to cut twill. The thickest you're supposed to cut on it is .3mm (.012"), and twill, especially with a backer, is much thicker than this.
I don't know if it had enough downforce or power to cut twill, either. I can't seem to find that info, but I'm guessing with it's slow cutting speeds that it might get bogged down with twill.
The Graphtec WILL cut it, and from what I've been told, it won't void your warranty to cut it, even tho it's not specifically mentioned in what it can cut. But you might want to check with Graphtec about that for sure.
Graphtec America: Cutting Plotters, Vinyl Cutters, InkJet Printers, Wide Format Image Scanners, Data Loggers, Acquisition platforms, Electronic Testing Instruments, Craft Robo
Good luck!