Paul,
I'd rather see you use a Dremel tool with a friction cutting disc. Using very light pressure, you use heat to melt away the acrylic rather then a mechanical means to cut it. Tape the area off (on both sides), including the cutting path, with masking tape, draw the pattern you want to cut-out on the tape, and fire up the Dremel. I think you will also be better off making several passes. Finish off the cut out with some sandpaper.