Cutting apart a striped ball shape in sep shapes

Hey there. Can someone please help me with how to cut apart a striped ball or balloon shape into separate shapes.  I've attached a sample file.  This is being sent to a laser cutter and I need to adjust for kerf, so I need the 7 curved shapes to be on their own.  I can't use the knife tool to create this because I need the curves the way I need them, not my wonky knife curve, so I had to create it with the outside balloon shape then put in the curved lines.  Is my only option to copy and paste the curved lines on top of each other, then cut the balloon wall where they intersect, join the paths, then do the same for the next shape down?

Thanks.