Problem is that your material is nothing like the vinyl material used in vinyl cutters.
The material used for vinyl sign cutting has backing sheet and the swivel head cutter is adjusted to only cut the vinyl this is to allow the machine to work. The cutter head on the machine only moves in the x direction the y direction is controlled by moving the whole lot backwards and forwards, this method cannot be used if you want to cut right through the material. You will need some form of gantry machine and the smallest radius cut will be limited, probably best to consider a laser but that will put the price up.

Peter