When doing long straight cuts like I'll break the stripe down into a bunch of small lines about 1' long each.
Here are the steps:
1) Make the stripes with your rectangle tool
2) Convert it to a path
3) Edit path
4) Cut a bunch of breaks in the rectangle about a foot long.
5) line up all the points to touch each other so it makes the shape.
6) cut
So really your cutting a bunch of line segments that make up the stripe. But the cutter will only have to track the length of the material once, reducing the chance of going off track.
On screen you won't be able to give it a fill because its all a bunch of open ended paths touching each other...
Hope this helps.