In V-carve, even with flip machining, you will have to make this 2 separate jobs.
Be sure to home the machine to the top of the stock piece, rather than the bed of the machine.
Drill a small hole where you want the center of the chair to be so that you can align the X and Y axes exactly on both sides when you flip.
If you want to cut the stool out, you will need to hand-draw a vector circle in vcarve that is the same diameter of this stool’s circle. You can add tabs to that vector circle so that the stool does not dislodge during the cut.