The only thing I can think of is that the circle $\mathbb{T}$ in its discrete topology is isomorphic to $\mathbb{R} \oplus \mathbb{Q}/\mathbb{Z}$ in its discrete topology, and the Pontryagin dual of this will be a continuum power of the Pontryagin dual $\mathbb{Q}_\mathrm{disc}^\prime$ (as you were just describing), times $\mathbb{Q}/\mathbb{Z}_\mathrm{disc}^\prime$. The latter can be identified with the product of all additive groups of $p$-adic integers $\mathbb{Z}_p$, with $p$ ranging over all primes. I somehow doubt one can do any better than that.