  We calculate the small-momentum expansion of vector, axial-vector, scalar,
and pseudo-scalar heavy-quark current correlators in the large-$\beta_0$ limit
of QCD, extending the analysis of Grozin and Sturm beyond the vector current.
Our results are used to study the higher-order behaviour of dimensionless
ratios of vector and pseudo-scalar moments used for the precise extraction of
the strong coupling, $\alpha_s$, from relativistic quarkonium sum rules and
lattice data, respectively. We show that these ratios benefit from a partial
cancellation of the leading renormalon singularities. Our results can guide the
design of combinations of moments with improved perturbative behaviour.
