  We use the newly developed technique of inverse quantum hamiltonian reduction
to investigate the representation theory of the simple affine vertex algebra
$\mathsf{A}_{2}(\mathsf{u},2)$ associated to $\mathfrak{sl}_{3}$ at level
$\mathsf{k} = -3+\frac{\mathsf{u}}{2}$, for $\mathsf{u}\ge3$ odd. Starting from
the irreducible modules of the corresponding simple Bershadsky-Polyakov vertex
operator algebras, we show that inverse reduction constructs all irreducible
lower-bounded weight $\mathsf{A}_{2}(\mathsf{u},2)$-modules. This proceeds by
first constructing a complete set of coherent families of fully relaxed
highest-weight $\mathsf{A}_{2}(\mathsf{u},2)$-modules and then noting that the
reducible members of these families degenerate to give all remaining
irreducibles. Using this fully relaxed construction and the degenerations, we
deduce modular S-transforms for certain natural generalised characters of these
irreducibles and their spectral flows. With this modular data in hand, we
verify that the (conjectural) standard Verlinde formula predicts Grothendieck
fusion rules with nonnegative-integer multiplicities.
