  Cuspidal automorphic representations $\tau$ of $\mathrm{PGL}_2$ correspond to
global long root $A$-parameters for $\mathsf{G}_2$. Using an exceptional theta
lift between $\mathrm{PU}_3$ and $\mathsf{G}_2$, we construct the associated
global $A$-packet and prove the Arthur multiplicity formula for these
representations when $\tau$ is dihedral and satisfies some technical
hypotheses. We also prove that this subspace of the discrete automorphic
spectrum forms a full near equivalence class. Our construction yields new
examples of quaternionic modular forms on $\mathsf{G}_2$.
