Do you need to cut from the middle of the piece or just take slices off?
I'm thinking a band saw might be a good choice but finding one that can use small blades to cut the curves might be a challenge. The old Inca band saws were advertised the take a 1/16" blade but good luck finding the saw or the blades.
Can you post more details about your project or maybe some pictures?
A scroll saw could cut the wood but I think it's going to be really, REALLY slow going. Maybe one of the premium machines could do it? Before you go that route try to find a scroll sawing forum and see if anyone has a recommendations.