We report on the anomalous magnetization dynamics of the cycloidally modulated spin textures under the influence of uniaxial anisotropy in multiferroic GaV4S8. The temperature and field dependence of the linear ac susceptibility [χ1ω′ (T,H)], AC magnetic loss [χ1ω′′ (T,H)], and nonlinear AC magnetic response [M3ω (T,H)] are examined across the magnetic phase diagram in the frequency range f=10-10000Hz. According to recent theory, skyrmion vortices under axial crystal symmetry are confined along specific orientations, resulting in enhanced robustness against oblique magnetic fields and altered spin dynamics. We characterize the magnetic response of each spin texture and find that the dynamic rigidity of the Néel skyrmion lattice appears enhanced compared to Bloch-type skyrmions in cubic systems, even in the multidomain state. Anomalous M3ω and strong dissipation emerge over the same phase regime where strong variations in the cycloid pitch were observed on lowering temperature in recent small-angle neutron-scattering experiments [White, Phys. Rev. B 97, 020401(R) (2018)10.1103/PhysRevB.97.020401]. Here, we show that strong anisotropy also drives an extended crossover of the zero-field cycloid texture in GaV4S8. The frequency dependence of these dynamic signatures is consistent with that of a robust anharmonic spin texture exhibiting a correlated domain arrangement. The results underpin the essential role of magnetic anisotropy in enhancing the rigidity of topological spin textures for diverse applications.