Updated design. Took out a stray path off the artboard that was stuck in the original.
There are several ways of dealing with this file. If you want different colors for the pieces, you can keep the pieces separate and have their paths colored differently and just cut the pieces you need from the different wood. But in the end the most efficient use would be to cut three sets of pieces from three different kinds of wood and then mix and match them. Once again, a fourth color of Proofgrade plywood would make a difference so we could solve the four color map problem and get a different colored edge.
Since there are only three types of Proofgrade, this simplifies the design greatly. I went ahead and removed all the double lines, including the inner line of the frame. That gets cut by the lines of the pieces. So You have only three operations. The one set of pieces and frame. The base and then the engrave.
I rotated 45 degrees to save material. That extends the engrave time a little, but it only took 5 minutes to do the whole thing.