Recently, methanol was identified as a sensitive target system to probe variations of the proton-to-electron mass ratio $mu$ [Jansen emph{et al.} Phys. Rev. Lett. textbf{106}, 100801 (2011)]. The high sensitivity of methanol originates from the interplay between overall rotation and hindered internal rotation of the molecule -- i.e. transitions that convert internal rotation energy into overall rotation energy, or vice versa, give rise to an enhancement of the sensitivity coefficient, $K_{mu}$. As internal rotation is a common phenomenon in polyatomic molecules, it is likely that other molecules display similar or even larger effects. In this paper we generalize the concepts that form the foundation of the high sensitivity in methanol and use this to construct an approximate model which allows to estimate the sensitivities of transitions in internal rotor molecules with $C_{3v}$ symmetry, without performing a full calculation of energy levels. We find that a reliable estimate of transition sensitivities can be obtained from the three rotational constants ($A$, $B$, and $C$) and three torsional constants ($F$, $V_3$ and $rho$). This model is verified by comparing obtained sensitivities for methanol, acetaldehyde, acetamide, methyl formate and acetic acid with a full analysis of the molecular Hamiltonian. From the molecules considered, methanol appears to be the most suitable candidate for laboratory and cosmological tests searching for a possible variation of $mu$.
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