How to remove overlapping paths for laser cutting

Hi there.

Thanks to the forum I have been able to do the design to the left, where each stripe of the balloon shape is it's own shape.  They can be cut on a laser cutter then put together again like a puzzle.   Now I am creating a laser cut design with overlapped shapes, and I need to remove the paths in the overlapping parts so that the final cuts would look like the picture on the right, where I simply used fills to hide the overlaps.  I have tried all of the path tools, plus shape builders and I can't seem to cut out those stripes from the lower balloon in the overlap.  I managed to do it once, but I worked on it for hours and it was a fluke I can't repeat;) 

I need all of these separate shapes (including the truncated stripes) so I can enlarge each piece to account for kerf on the cutter.

Any help would be greatly appreciated!

Thanks

Sue