Recently, chirped and tilted fiber Bragg gratings (CTFBGs) have received great attention because they can realize suppression of stimulated Raman scattering (SRS) in high-power fiber lasers. In this study, the possible coupling between the core modes and cladding modes in CTFBGs inscribed in large-mode-area double-cladding fibers is investigated for the first time. Theoretical results show that the coupling between the LP11 mode and cladding modes would destroy the transmission spectra envelope only considering the coupling of LP01 for single-mode CTFBGs, which will degenerate the SRS suppression performance. This was confirmed experimentally by measuring the spectral response under different mode excitations. A reliable method is demonstrated to ease the LP11-excitation-induced spectral deterioration by choosing an appropriate chirp rate for the inscription of CTFBGs, which is useful for improving the Raman suppression effect of large-mode-area double-cladding CTFBGs in high-power fiber lasers.