In this study, we investigate a (3+1)-dimensional generalized nonlinear wave equation modeling bubbly liquid systems using the non-classical symmetry method. Distinct classes of symmetries are identified and systematically categorized, leading to a variety of novel and exact solutions. Beyond the analytical results, we conduct a comprehensive dynamical analysis, including bifurcation structures, equilibrium points, and chaotic behavior. Chaos diagnostics are carried out using phase portraits, Poincaré maps, power spectra, Lyapunov exponents, bifurcation diagrams, and return maps. The novelty of this study lies in deriving previously unreported families of exact solutions for the (3+1)-dimensional bubbly liquid wave equation through non-classical symmetries, and in coupling these analytical findings with a systematic bifurcation and chaos analysis. This combined framework provides new insights into the nonlinear dynamics of bubbly liquids that have not been addressed in earlier studies. The findings reveal new families of exact solutions and complex nonlinear behaviors, offering valuable insights into the transition mechanisms between ordered and chaotic phases in wave propagation within bubbly liquids.