Much easier to bend the wood than to try and cut out a curved shape. Get some thin wood strip like the kind they use on model ships for planking and trim, soak it in a glass of water overnight, then next day take it out, form it to the curve of the door cutout by attaching it to the door with masking tape. Once it dries, it'll spring out a little, but not much. Then just test fit, trim the excess off the ends, stain, varnish or clear acrylic, and CA glue into place. You'll get a perfect fit around the cutout radius without trying to saw out a curved piece that's not only got the correct curve, but also has uniform thickness throughout. It's MUCH easier to use the strip/soak/bend method.
BTW... if you have a Hobby Lobby nearby, they have a nice selection of balsa and basswood in different thicknesses. You can stain it any color you like to make it look like "walnut," "cherry," or whatever.