That is an articularly nice living hinge design! 
making round things was the first challenge I went to when I first got my Glowforge. I did not understand about living hinges but have an oak cane bent at the top and so went off in that direction.
I did find that a deep engrave really stabilized the round, the ability to expand or contract the circle really helps as if the wood is solid any change in temperature or humidity really changes the diameter.
Also what I discovered was that a locking pattern could hold the ends together though it looks like you are just matching the design at the ends which could be why keeping it from trying to be an exclamation point takes some effort. Also, I have noted that a curved area of stress has a lot less stress than if the stress is focused to a point.
This is one example of a whole line of lamps and then I also tried making the living hinge on the inside only.