It looks to me like you are just asking the material to hold a level of detail it is not capable of performing. Also consider that wood is not a solid and consistent material. If you look at the grain direction the thin details in your photo are opposite the grain, so those thin walls are also (roughly) in the wood’s weakest direction.
Really dense wood, and maybe bamboo could be pushed to maybe 1/16 thick. But any tool deflection, or flaw in the wood and it would break. Even if it survives the CNC process it could break apart later (age, wood expansion-contraction, etc).
Warning: This is a slippery slope (but also fun). The overall shape could be pocketed out of the wood, then fill the pocket with resin. Once the resin has hardened you could then CNC the detail you are looking for.