Molecular overtone transitions provide optical frequency transitions sensitive to variation in the proton-to-electron mass ratio (μ = mp/me). 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 μ attainable using TeH+. We find that laser cooling of TeH+ can lead to significant improvements on current μ variation limits.