The expanding realm of compliant and flexible soft solids is marked by notable advantages, offering secure interactions with humans and delicate objects. This study employs advanced numerical methodologies to meticulously investigate the propagation of elastic plate waves within a fiber-reinforced prestretched compressible material, characterized by the Gent hyperelasticity model. The formulation encompasses the elastic tensor and underlying wave equations in Lagrangian space, integrating the theory of nonlinear elasticity with linearized incremental equations. To compute the dispersion characteristics of two fundamental guided wave modes, an extension of the Semi-Analytical Finite Element (SAFE) method is introduced. The predictive capabilities of this numerical framework undergo rigorous validation utilizing previously published data for hyperelastic material models. Subsequently, the investigation delves into the intricate effects of applied prestretch, reinforced fiber orientation, and material parameters on the dispersion characteristics of fundamental Lamb modes. A limiting case of the neo-Hookean material model is scrutinized to elucidate implicit dependencies, with a specific emphasis on the strain-stiffening effect captured through the Gent material model. The study's findings unveil that the manipulation of material anisotropy and fiber direction provides precise control over anti-symmetric and symmetric modes. Moreover, a critical revelation emerges as the study identifies the existence of a threshold prestretch, inducing a snap-through instability in Gent-type materials. This pivotal revelation leads to a consequential alteration in the dispersion characteristics of fundamental Lamb modes. The comprehensive exploration and insights presented in this study significantly contribute to the understanding and control of wave propagation in hyperelastic materials, particularly those exhibiting Gent-type behavior.