Molecules with deep vibrational potential wells provide optical intervals sensitive to variation in the proton-electron mass ratio ($mu$). On one hand, polar molecules are of interest since optical state preparation techniques have been demonstrated for such species. On the other hand, it might be assumed that polar species are unfavorable candidates, because typical molecule-frame dipole moments reduce vibrational state lifetimes and cause large polarizabilities and associated Stark shifts. Here, we consider single-photon spectroscopy on a vibrational overtone transition of the polar species TeH$^+$, which is of practical interest because its diagonal Franck-Condon factors should allow rapid state preparation by optical pumping. We point out that all but the ground rotational state obtains a vanishing low-frequency scalar polarizability from coupling with adjacent rotational states, because of a fortuitous relationship between rigid rotor spacings and dipole matrix elements. We project that for good choices of spectroscopy states, demonstrated levels of field control should make possible uncertainties of order $1 times 10^{-18}$, similar to those of leading atomic ion clocks. The moderately long lived vibrational states of TeH$^+$ make possible a frequency uncertainty approaching $1 times 10^{-17}$ with one day of averaging for a single trapped ion. Observation over one year could probe for variation of $mu$ with a sensitivity approaching the $1 times 10^{-18}/textrm{yr}$ level.
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