Inspired by recent experimental work, we study the control over the laser-driven dissociation of the HeH+ ion in the electronic ground state. Shaped pulses with peak intensities below 1012 W cm-2 are obtained by phase modulation of high-intensity transform-limited femtosecond pulses. We investigate the performance of pulse shaping for a number of shaping parameters targeting both vibrational and rotational excitation pathways. The numerical results show that pulse shaping is most effective at low pulse energies and broad spectral bandwidths, while intense transform-limited pulses with narrow spectral bandwidths maximize dissociation. We show that the control achieved with a quadratic chirped pulse optimized for vibrational ladder climbing, a cascade excitation process of adjacent vibrational levels, is hindered by rotational motion leading to significantly reduced dissociation. Moreover, pulse shaping using higher-order polynomial phase functions is found to provide only a marginal increase in dissociation yields. Our results provide additional insights into the coherent control of bond breaking in diatomic molecules, and demonstrate the efficacy of pulse shaping for a range of pulse energies.