Lasers designed to emit at multiple and controllable modes, or multi-wavelength lasers, have the potential to become key building blocks for future microwave photonic technologies. While many interesting schemes relying on optical injection have been proposed, the nonlinear mode coupling between different modes of a multi-wavelength laser and their dynamical behavior under optical injection remains vastly unexplored. Here, we experimentally and numerically study the effect of optical injection around the suppressed mode of a dualwavelength laser and the resulting interactions with the dominant mode. We highlight a wavelength shift of the dominant mode triggered by injection locking of the suppressed mode and report a strong impact of the mode suppression ratio on the locking range. Finally, we show numerically that the cross-coupling parameter between the two modes might have a key role in this effect.