Thought I'd provide an update . . .
Its been weeks, I ran into a ton of issues trying to cut the material clean.
I tried several thicker materials in a attempt to cut to the fiber and maintain the original thickness. Each attempt was extremely time consuming and resulted in failure for various reasons. Almost defeated I decided to go back to my original material choice and only make low cuts to give the appearance of the raised pattern.
Below are the results (original on the LH side and replacement piece on the RH side.