Knowledge of the time-dependent behavior of polyvinyl alcohol (PVA) gel polymer electrolytes (GPEs) is essential for their application in structural supercapacitors (SSCs) at lower degrees of integration (DoI). Due to the soft, viscoelastic nature of GPEs, their mechanical response under compressive and transient loading is critical for maintaining necessary long-term electrode contact. To enable reliable structural design configurations, a tailored PVA GPE, prepared via a freeze–thaw method, was examined using compressive stress-relaxation testing. The mechanical response was modeled using a novel, non-linear viscoelastic constitutive framework, which couples a time-dependent elastic modulus for the non-linear loading phase with a Prony series for accurate relaxation prediction. The parameters for this practical framework were successfully derived from one single stress-relaxation experiment. Experimental results confirmed pronounced viscoelastic relaxation and up to 10% cyclic hardening. Implemented via finite-element method (FEM) analysis, the non-linear model achieved high accuracy (0.8% average relative deviation), significantly outperforming a linearized model (1.36% deviation). This validated framework is crucial for optimizing the mechanical stability of SSC assemblies and predicting the GPE’s short-time response under transient loading events.