I think what I would try is import that like mentioned above but into Pro/E or Solidworks sheetmetal package. If you like I could try it for you. email@example.com
maybe if it does not import properly we would want to retrace the flat using illustrator. We do this a lot in Maya. Place a 2d image from photoshop onto a model… get the image perfect and retrace it in illustrator to get the flat pattern for a complex compound surface. We call that ‘maya softgoods’ works well… but we don’t take it to sheetmetal like your idea.
what type of laser cutter do you have? I use a universal laser systems x-660, you will want to do what is stated above and then play with it in Illustrator. The only way I see you getting your final result to look like the image you showed is to use acrylic.