What I would be inclined to do (NOT myself!) is find someone that could use a router to remove a deep layer of wood from the back of the guitar (mainly behind the bridge), leaving a "ledge" around the edges, then cutting a thin piece of wood to fit the cutout. It might be a bit trial and error to get to the amount of wood to remove and get the right balance and not affect the tone too severely.
The back might end up something like this (OR you could try to get a matching wood). That would preserve the rounded edges of the guitar and avoid the need to then have to match the finish. (This example has the back extending to the edges, part of the original build, rather than inset after the fact).