Interesting problem...
If you want a reasonably accurate shape and are willing to build a jig, I can envision a pretty accurate way to rough cut it on the table saw that would then be smoothed as others have indicated.
The attached shows a very simplified concept. The jig consists of a rotating hinged board onto which the workpiece is attached maybe via double sided tape or screws. The holding knob/mechanism that locks the angle is not shown. The start position is with the workpiece vertical and just against the saw blade. The fence is moved away from the blade a defined distance and the workpiece is rotated a defined angle - this might best be done with an electronic angle finder. To be clear, this is done with the saw stopped. The piece is ripped and the process repeated until the piece is 90 degrees. The other (upper) half of the board is done by flipping the workpiece over and repeating the process.
Since both the shape and tangent of an ellipse are defined by simple equations, the angles and offsets are easily calculated in a spreadsheet. Dont forget that the origin of the workpiece is offset in the x dimension by the thickness of the backer board and, depending on hinge type, the hinge thickness. This affects both x and y movement as the piece is rotated.
PM if you want to discuss any of this.