We investigate the ground-state structures of a quasi-two-dimensional Rydberg-dressed spin-1 Bose gas with SU(3) spin-orbit coupling confined in a harmonic trap. The mean-field energy functional contains contact interactions, component-dependent nonlocal soft-core Rydberg interactions, and a momentum-dependent SU(3) coupling generated by Gell-Mann matrices. The ground states are obtained by imaginary-time evolution, while the nonlocal interaction term is evaluated in momentum space using the convolution theorem and Meijer's G-function. By treating the intraspecies and interspecies Rydberg interaction strengths as independent control parameters, we identify how their relative strength regulates the spatial organization of the three spin components. When the interspecies Rydberg repulsion is relatively strong, the system tends to reduce the overlap between different spin components and develops a radial component-separation structure. In contrast, increasing the intraspecies interaction can enhance the overlap among the three components and drive the density profile toward a lattice-like distribution. To quantify these changes, we introduce a normalized component-overlap integral and a radial-separation measure, which directly characterize the evolution from radial separation to enhanced component mixing and the reverse trend induced by stronger interspecies repulsion. The SU(3) spin-orbit coupling further locks the relative phases among the three internal states and converts the Rydberg-induced component rearrangement into correlated phase patterns and spin textures. As a result, ordered spin-density distributions and spin-vortex textures can be generated by tuning the relative strengths of the intraspecies and interspecies Rydberg interactions. These results show that asymmetric Rydberg interactions provide an additional degree of freedom for manipulating density modulation, component separation, phase organization, and spin textures in multicomponent spin-orbit-coupled Bose gases.