The fastest way I can imagine to do this is to take a photo along the long axis of the board, import that into your 3D software, and then bend the type layout to match the curve in the photo. It is very quick and dirty, but may be effective enough.
I have a digitizing probe and you could use a large grid (maybe 2"x2")to get a basic representation of a surface quickly, but could probably not hit your $70 price point for a one-off design, due to the time required for the 3D scan.
A third possibility would be a 2d scan. In a similar fashion to the photo method, which would give you a rough guide of the curve on a single axis, you can do a single pass probe scan down the board. Instead of a grid, you would probe a straight line of sample points down the long axis of the board, generating a curved line the shape of the board.