Edit: my previous answer was incorrect
No, this one you cannot do. If one had such an arrangement of circles in the solid torus inside $R^3$, and quotiented by a rotation, then by the equivariant Dehn's lemma there would be a collection of essential disks in the complement of the split link that are invariant under the symmetry.
Since the link components get permuted, no disk can have a fixed point of the cyclic action. So they are disjoint from the axis of symmetry of the rotation. Hence the components are unknotted in the solid torus, a contradiction
