A novel FFT-based computational framework for coupling non-local elastic-plastic damage model is proposed in this paper, which is employed to accurately simulate the transverse tensile behaviors of unidirectional (UD) CF/PEEK composite materials under different temperatures. To address the distinct material properties exhibited by different microscopic constituents within the composite, two approaches are employed: (i) an integral-type regularized traction-separation damage model is applied to the interphase debonding, and (ii) an implicit gradient regularization technique for Lemaitre-type damage (matrix cracking) of the PEEK. By coupling the non-local damage model capable of accurately characterizing different failure modes within composites with an FFT computational framework, accurate predictions of the transverse tensile performance of UD-CF/PEEK composites at different temperature conditions can be achieved. Furthermore, the influence of non-local feature parameters and interphase mechanical properties on the composite's mechanical behavior is thoroughly discussed. Comparisons with experimental results affirm the accurate prediction of the transverse tensile behavior of UD-CF/PEEK composites through the proposed computational framework. Ultimately, based on this framework, the transverse tensile modulus and strength of UD-CF/PEEK composites, considering different fiber volume fractions and temperatures, are predicted, successfully demonstrating the effectiveness and applicability of the proposed approach in forecasting the mechanical behavior of composites.