We develop a modified fast Fourier-Galerkin method tailored for 3D boundary integral equations on torus-diffeomorphic surfaces, addressing challenges posed by a non-square-integrable, weakly singular quaternary kernel function. By applying a shear transformation and fixing certain variables, we identify a uniform analytical continuation radius that informs a strategic truncation of the matrix representation. This allows for the sparse approximation of the 3D boundary integral operators, maintaining high accuracy. The paper outlines the method's formulation, its theoretical underpinnings, stability, and convergence analyses, and validates the approach through numerical experiments, demonstrating improved efficiency and applicability in complex geometrical configurations.