The correct form for the Green's function for the vector Helmholtz equation is
I do not intend to prove a key element of this assertion (that the products of the involved reduce to Legendre polynomials in the angle between the arguments times the identity tensor) in class. Instead, I leave it as an exercise. To get you started, consider how similar completeness/addition theorems are proven for the spherical harmonics themselves from the given orthonormality relation.
With these relations in hand, we end our mathematical digression into vector spherical harmonics and the Hansen solutions and return to the land of multipolar radiation.