A Weyl semimetal with strong electron-phonon interaction can show axionic coupling in its insulator state at low temperatures, owing to the formation of a charge density wave (CDW). Such a CDW emerges in the linear chain compound Weyl semimetal Ta₂Se₈I below 263 K, resulting in the appearance of the dynamical condensed-matter axion quasiparticle. In this study, we demonstrate that the interchain coupling in Ta₂Se₈I can be varied to suppress the CDW formation with pressure, while retaining the Weyl semimetal phase at high temperatures. Above 17 GPa, the Weyl semimetal phase does not survive and we induce superconductivity, due to the amorphization of the iodine sub-lattice. Structurally, the one-dimensional Ta-Se-chains remain intact and provide a superconducting channel in one dimension. We highlight that our results show a near-complete suppression of the gap induced by the axionic charge-density wave at pressures inaccessible to previous studies. Including this CDW phase, our experiments and theoretical predictions and analysis reveal the complete topological phase diagram of Ta₂Se₈I and its relationship to the nearby superconducting state. The results demonstrate Ta₂Se₈I to be a distinctively versatile platform for exploring correlated topological states.