  Vibrational overtones in deeply bound molecules are sensitive probes for
variation of the proton-to-electron mass ratio $\mu$. In nonpolar molecules,
these overtones may be driven as two-photon transitions. Here, we present
procedures for experiments with ${\rm O}_2^+$, including state-preparation
through photoionization, a two-photon probe, and detection. We calculate
transition dipole moments between all $X\,^2\Pi_g$ vibrational levels and those
of the $A\,^2\Pi_u$ excited electronic state. Using these dipole moments, we
calculate two-photon transition rates and AC-Stark-shift systematics for the
overtones. We estimate other systematic effects and statistical precision.
Two-photon vibrational transitions in ${\rm O}_2^+$ provide multiple routes to
improved searches for $\mu$ variation.
