Transition metal oxides (TMO) are an ideal playground for the study of electronic correlations in solids. The Mott metal-insulator transition (MIT) is a dramatic manifestation of these correlations which occurs in several TMO families including titanates and vanadates. Typical control parameters of the MIT are the bandwidth and electronic doping, which can be experimentally varied as a function pressure and heterovalent cationic substitutions, respectively. In practice, MIT studies are hampered by the occurrence of structural phase transitions driven by a strong interplay between electronic and lattice degrees of freedom or by the disorder inherent of chemically substituted systems. In order to overcome these difficulties, we propose the quadruple perovskite NaMn₇O₁₂ (NMO) as a model system where only the electronic degrees of freedom drive the pressure-controlled MIT observed at ~17GPa. The reason is the very high density of the cubic I_m-3 room temperature structure, which is due to an unusually large tilt of the MnO₆ octahedra. By considering all possible distortions in perovskite-like structures, we expected that under high pressure, the above symmetry is stable against distortions, thus effectively freezing the lattice degrees of freedom. Our high-pressure Raman study confirms this expectation, with the mode frequencies increasing continuously with hydrostatic pressure up to 22 GPa, without any anomaly that may be related to structural phase transitions. Additionally, the line shape of the lowest frequency mode with T_g symmetry is found to be very sensitive to pressure. Our analysis of the lineshape using a Briet-Wigner-Fano model enabled us to estimate the phonon-carrier coupling, which is proportional to the carrier density, n. These results show that n increases up to 11 GPa and then levels off, thus providing a qualitative indication of the evolution of the electrical transport under pressure. Complementarily, we measured the infrared-active phonon modes as a function of pressure, which enabled us follow the evolution of the structure across the MIT.