<p>Frictional contact problems remain a critical challenge in computational mechanics. In recent years, polygonal elements have provided a more flexible meshing scheme for contact problem calculations. In this work, we propose a robust computational algorithm by constructing a polynomial projection for linear-compatible strain fields, extending polytopal composite elements to finite-strain hyperelasticity by the utilization of least squares approximation. The framework is applied to frictional contact problems involving large deformations. For contact modeling, the bi-potential method applies the augmented Lagrangian method to contact laws, resulting in an implicit projection equation on the Coulomb friction cone. This can eliminates artificially defined parameters to implicitly determine contact forces, avoiding the construction of contact stiffness matrices. Numerical experiments demonstrate stable accuracy and consistent convergence, establishing a new reference for polygonal finite element applications in frictional contact problems.</p>