We investigate predissociation in homonuclear (87Rb87Rb+) and heteronuclear (87Rb7Li+) long-range Rydberg atom-ion molecules. Owing to their micron-scale bond lengths, these dissociate on time scales far removed from those of more tightly bound diatomic molecules. We employ the eigenchannel R-matrix method to compute predissociation rates for a broad range of principal quantum numbers n. The rates depend strongly on the mass, but more remarkably display a rapid and periodic variation as a function of n as well as within a single vibrational ladder. A semiclassical Landau-Zener-Stückelberg analysis reveals that Stückelberg interference governs the decay process and produces the observed variation in the molecular lifetime. Although the heavy mass of the homonuclear 87Rb87Rb+ molecule constrains its predissociation rates to a sub-kHz level, the lighter molecule 87Rb7Li+ dissociates on time scales competitive with radiative and collisional decay. This can enable in situ study of non-adiabatic decay via ion microscopy.