This work introduces a computationally efficient, second-gradient decoupled homogenization framework for hyperelastic fiber-reinforced composites under large deformations, specifically designed to handle extreme stiffness contrasts. Classical first-order computational homogenization fails to capture microstructure-dependent bending stiffness \cite{madeo}. A critical limitation is its inability to accurately model fiber buckling under flexural loading at high contrast (e.g., $c = 3000$); while adequate for tensile responses, it is deficient for compression-induced instabilities in stiff-fiber/soft-matrix composite materials \cite{gamra2026}. To address this, we first establish a direct implementation using \textbf{$C^1$} B-spline elements to discretize the second-gradient energy, which yields accurate results with a marginal online cost increase but imposes restrictive continuity. We then develop a novel micromorphic reformulation of the theory. By introducing a micromorphic variable kinematically linked to the deformation gradient and enforcing this link via Lagrange multipliers, the framework bypasses the need for $C^1$ continuity, allowing the use of standard $C^0$ finite elements. However, the higher-order homogenisation is achieved while maintaining a classical first-order Cauchy continuum description at the micro-scale. In this context, the fibres are modelled as quasi-inextensible, hyperelastic solids, and the matrix as quasi-incompressible solids. The methodology employs a pre-identification process of the second gradient hyperelastic model parameters from an offline RVE simulation, including both the first and second deformation gradient and their Work-Conjugate stresses , the Cauchy stress, and double stress tensor. Numerical tests have been used to validate the framework. The results show that both implementations significantly outperform first-order homogenisation across a wide range of stiffness contrasts ($E_f/E_m$ from 10 to 10000). For instance, in the case of transverse bending, the first-order model error increases to 37\%, while the second-gradient models maintain errors below 8\%, successfully capturing the correct bending stiffness and buckling response. The study concludes that the micromorphic decoupled scheme offers an efficient, accurate, and practically implementable multiscale strategy for composites where extreme anisotropy and structural instabilities are critical.