The reason I'm thinking a CNC router is that I could only replace the burl veneer and leave the 2" frame at the outside. The edges and corners of the frame are where I noticed it's not flat.
The other thing is one could actually make the top flat with the router, using the lowest point as a reference and routing the rest to the same height. Then as a second operation, route the area for the veneer. Probably using 3 of the 4 corners to form the plane for the router would work since the corners are probably lower than anyplace else.