I'm really surprised there is not a router plate mounted feeler bearing on an adjustable slide. This would allow me to set the bearing a certain distance from the workpiece, and then follow the perimeter of the workpiece while the piece cuts from underneath.
The only way this would work is if there was a way to keep the line between the bearing and the bit perpendicular to the edge of the workpiece, which would be difficult with a curved edge.
Your piece looks fairly large but not that complex. What I would do is to make a template that is the size of the inner edge of your desired groove. You can spend as much time as needed to refine the template until it is the exact size and shape you need. Then, attach the template to the workpiece (double-sided tape?) and rout around that. You would still need to be really careful to keep the bearing against the template at all times.