This paper presents a classical mechanical analysis of a truly nonlinear oscillator with position-dependent mass (PDM). Starting from the Hamiltonian formulation, the corresponding Lagrange equation of motion is derived, yielding a strongly nonlinear Liénard-type oscillator. An approximate analytical solution is proposed using the cosine Ateb function, and a first integral of energy type is identified, where both kinetic and potential energies are expressed as products involving the mass function. A frequency estimation method based on He’s formalism is developed to explore the interaction between PDM parameters and the nonlinear coefficients. The theoretical approach is applied to three representative mass distributions: symmetric and asymmetric exponential forms, and a polynomial-type function. Time histories and phase portraits are plotted to examine the dynamic response. It is shown that the motion remains stable and periodic across various conditions. These findings have potential applications in structural dynamics, molecular physics, semiconductors, and quantum systems.