We design fast trajectories of a trap to transport two ions using a shortcut-to-adiabaticity technique based on invariants. The effects of anharmonicity are analyzed first perturbatively, with an approximate, single relative-motion mode, description. Then, we use classical calculations and full quantum calculations. This allows us to identify discrete transport times that minimize excitation in the presence of anharmonicity. An even better strategy to suppress the effects of anharmonicity in a continuous range of transport times is to modify the trajectory using an effective trap frequency shifted with respect to the actual frequency by the coupling between relative and center-of-mass motions.