  The family of $A^{(1)}_2$ models on the square lattice includes a dilute loop
model, a $15$-vertex model and, at roots of unity, a family of RSOS models. The
fused transfer matrices of the general loop and vertex models are shown to
satisfy $s\ell(3)$-type fusion hierarchies. We use these to derive explicit
$T$- and $Y$-systems of functional equations. At roots of unity, we further
derive closure identities for the functional relations and show that the
universal $Y$-system closes finitely. The $A^{(1)}_2$ RSOS models are shown to
satisfy the same functional and closure identities but with finite truncation.
