In this paper, we developed a numerical scheme based on barycentric rational interpolation to solve the generalized modified Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Using the proposed approach, we applied a direct linearization technique to transform the nonlinear generalized modified CH–DP equation into an equivalent linear formulation. Approximate solutions were constructed using barycentric rational interpolation basis functions. The spatiotemporal domain was discretized via barycentric rational interpolation, and a corresponding differentiation matrix was derived. Furthermore, theoretical analysis of error distribution and convergence properties was presented. Analytical results and numerical experiments validate the scheme's computational efficiency and high accuracy in solving the generalized modified CH–DP equation.