You could leave the carving as it is, just wood. But a swan, among many things - for example, dying in a ballet - is famous for being white...
Bleaching, which always works better on lighter woods, is far better than painting because you can still clearly see the wood grain. The problem is the yellow bill with its black knob and the black eyes, all set off by the white. What to do? Decisions. Decisions.
Whatever, do test on spare wood first.