  Molecular overtone transitions provide optical frequency transitions
sensitive to variation in the proton-to-electron mass ratio ($\mu\equiv
m_p/m_e$). However, robust molecular state preparation presents a challenge
critical for achieving high precision. Here, we characterize infrared and
optical-frequency broadband laser cooling schemes for TeH$^+$, a species with
multiple electronic transitions amenable to sustained laser control. Using rate
equations to simulate laser cooling population dynamics, we estimate the
fractional sensitivity to $\mu$ attainable using TeH$^+$. We find that laser
cooling of TeH$^+$ can lead to significant improvements on current $\mu$
variation limits.
