The appearance of unconventional superconductivity near heavy-fermion quantum critical points (QCPs) motivates investigation of pairing correlations close to a "beyond Landau" Kondo-destruction QCP. We focus on a two-Anderson-impurity cluster in which Kondo destruction is induced by a pseudogap in the conduction-electron density of states. Analysis via continuous-time quantum Monte Carlo and the numerical renormalization group reveals a previously unstudied QCP that both displays the critical-local moment fluctuations characteristic of Kondo destruction and leads to a strongly enhanced singlet-pairing susceptibility. Our results provide insights into the mechanism for superconductivity in quantum critical metals.