  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.
