  The Bershadsky-Polyakov algebras are the original examples of nonregular
W-algebras, obtained from the affine vertex operator algebras associated with
$\mathfrak{sl}_3$ by quantum hamiltonian reduction. In [arXiv:2007.03917], we
explored the representation theories of the simple quotients of these algebras
when the level $\mathsf{k}$ is nondegenerate-admissible. Here, we combine these
explorations with Adamovi\'{c}'s inverse quantum hamiltonian reduction functors
to study the modular properties of Bershadsky-Polyakov characters and deduce
the associated Grothendieck fusion rules. The results are not dissimilar to
those already known for the affine vertex operator algebras associated with
$\mathfrak{sl}_2$, except that the role of the Virasoro minimal models in the
latter is here played by the minimal models of Zamolodchikov's $\mathsf{W}_3$
algebras.
