  The Bershadsky--Polyakov algebras are the subregular quantum hamiltonian
reductions of the affine vertex operator algebras associated with
$\mathfrak{sl}_3$. In arXiv:2007.00396 [math.QA], we realised these algebras in
terms of the regular reduction, Zamolodchikov's W$_3$-algebra, and an isotropic
lattice vertex operator algebra. We also proved that a natural construction of
relaxed highest-weight Bershadsky--Polyakov modules gives modules that are
generically irreducible. Here, we prove that this construction, when combined
with spectral flow twists, gives a complete set of irreducible weight modules
whose weight spaces are finite-dimensional. This gives a simple independent
proof of the main classification theorem of arXiv:2007.03917 [math.RT] for
nondegenerate admissible levels and extends this classification to a category
of weight modules. We also deduce the classification for the nonadmissible
level $\mathsf{k}=-\frac{7}{3}$, which is new.
