Using quantum-mechanical, one-dimensional, non-Born-Oppenheimer simulations we study the control over the strong-field dynamics of the helium hydride molecular ion HeH+ due to interaction driven by short and strong two-color laser pulses. We calculate yields of two competing fragmentation channels: electron removal and dissociation. We find that by changing the relative phase of the two colors, we can select the dominating channel. Nuclear motion is decisive for explaining ionization in this target. Ionization yields are vastly underestimated when nuclear motion is excluded and they are substantially reduced in the heavier isotopologue HeD+. Coupling of the two lowest electronic states is crucial even for the ground-state dissociation process. © 2020 The Author(s). Published by IOP Publishing Ltd.