We study the dynamics of an optically injected quantum-dot laser accounting for excited states. Mapping of the bifurcations in the plane frequency detuning vs. injection strength shows that the relaxation rate scales the regions of locking and single- and double-period solutions, while the capture rate has a minor effect. Within the regions of time-periodic solutions, close to the saddle-node bifurcation boundary, we identify subregions where the output signal resembles excitable pulses as a result of the bottleneck phenomenon. We show that such emission is determined mainly by fluctuations in the occupation of the excited states. The interpulse time follows an inverse square root scaling law as a function of the detuning. In a deterministic system the pulses are periodic regardless of the detuning, but in the presence of noise, close to the locking region, the interpulse time follows a positively skewed normal distribution. For a fixed frequency detuning, increasing the noise strength can shift the mean of the interpulse time distribution and make the pulsations more periodic.