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No, this one you cannot do. If one had such an arrangement of circles in the solid torus, and quotiented by the cyclic symmetry, 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. All of these disks must meet the symmetry axis since the link is irreducible in the solid torus, an they can only intersect in a single point, since they are invariant under a cyclic group. But this implies that the linking number between each component and the solid torus is non-zero, a contradiction.