Is image trace the right tool for this task?

GOAL:  Re-create a photograph using 4-5 layers of paper cut out on a laser cutter, where each layer is a different shade of grey.  darkest layer on the bottom, then grey, then light grey, then white.

PARTIAL SOLUTION: Generate paths for the laser using illustrator to process the image.  I can get four levels of grey using image trace, but it's not quite what I need.  The top (white) layer path is fine, but the path of the next level of grey needs to encompass the white layer, but instead just traces around the grey.  If I could combine the white into next grey level path (combine them together into a single outline of the grey) that would be perfect.  But I don't know how to do that.

QUESTION: how do I combine the white and grey paths (that ajoin, but don't overlap) into a grey path that "consumes" the white path?  Such that the white path is no longer there.

EXAMPLE: Think of a radial gradient that is white in the middle and fades to black on the edges.  Image trace creates the four levels of grey.  There is a white circle path in the middle. The next grey path forms a ring, rather than a circle.  What I want is a path that just includes the outer edge of the grey, but not the inner edge.  When done the paths should form concentric shapes, like a topographic map where each level of grey encompasses the brighter levels above it.

So when the paper is cut with the laser, the profiles can be stacked and glued.

AQny suggestions greatly appreciated.

Matt