No experience and real help here but a few thoughts and questions.
1. Diameter and thickness of the plywood circles?
2. Not the iron on veneer that Joe didn't like?
3. Solid wood edge bending may be possible. Maybe steam bending 1/4" - 3/8" thick strips using the plywood circle as the home base bending form? A larger diameter is easier to bend because it has a gentler curve radius.
This finishes out at 12" diameter with a 1/4" thick edge band which is mitered at a 60 degree overlapping angle to finish.
Making it is a different matter!