This work presents a novel class of composite product-integration closed Newton--Cotes (CNC) integrators specifically designed for Hamiltonian systems governed by Caputo fractional derivatives of order $1 < \alpha < 2$. The proposed methodology establishes explicit full-history convolution weights while constructing a discrete variational principle within an extended phase space, ultimately yielding a symplectic mapping. Through rigorous analysis, we demonstrate global error bounds of $O(h^{2-\alpha})$ for CNC-2 and $O(h^{4-\alpha})$ for CNC-4 under conventional regularity assumptions. The linear stability characteristics are analyzed via the generating function of convolution weights, while long-term energy drift behavior is quantified. Additionally, efficient implementation strategies are developed that achieve $O(1)$ per-step computational complexity through systematic history truncation and sum-of-exponentials compression techniques. Comprehensive numerical experiments validate the theoretical predictions and demonstrate superior performance relative to existing fractional integrators.