Does the whole doll have to move?
What about having the body fixed and then putting the limbs and head on wires so they would wiggle about when the cake was moved? You might need to play about the the gauge of wire so that it is stiff enough to hold the pieces but not so stiff it doesn't move with a fairly small amount of movement of the cake.
If it's a single tier could you put it on an unstable base, if such a thing exists? Like a turntable but which would move on the horizontal axis too or put the cake board on a thick piece of sponge or something so the cake would moce about causing the doll to dance... or would that be asking for trouble?!