To unveil the influence of galaxy-galaxy interactions on the material transport driven by galactic bars toward the central regions of active galactic nucleus (AGN) galaxies, and to assess the efficiency of the combined mechanisms of interactions and bars in fueling massive black holes, we meticulously examine barred active galaxies in paired systems. Our study focuses on barred AGN galaxies in pairs with projected separations of $r_ p \,kpc $ and relative radial velocities of $ V< 500 \,km $ within $z<0.1$, identified by the Sloan Digital Sky Survey (SDSS). To quantify the impact of interactions on material transport by galactic bars, we also constructed a suitable control sample of barred active galaxies without paired companions, matched in redshift, absolute r-band magnitude, stellar mass, color, and stellar age distributions. Additionally, we calculated the structural characteristics of galactic bars through two-dimensional image modeling, considering that bars exhibit a wide range of shapes and sizes, which may influence their ability to channel material. From this study, we clearly found that nuclear activity (derived from the $Lum OIII $) increases as the projected separations between galaxy pair members decrease. Notably, barred AGN galaxies in close pairs ($r_p 25 kpc $) exhibit significantly higher nuclear activity compared to galaxies in the control sample. Additionally, barred galaxies with a close pair companion show enhanced nuclear activity across all ranges of luminosity, stellar mass, and color. We also found that barred AGN galaxies with longer bar structures exhibit more efficient nuclear activity compared to those with shorter bars. This trend is especially pronounced in barred AGN galaxies within close pair systems, which show a significant excess of high $Lum OIII $ values. Furthermore, we examined the central nuclear activity in barred AGNs undergoing major and minor interactions. Our findings show a clear escalation in nuclear activity as the pair projected separations decrease, particularly pronounced in major systems. Additionally, nuclear activity distributions in barred AGN samples within major and minor pairs exhibit similar trends. However, a significant deviation occurs among barred AGN galaxies in close pair systems within major interactions, showing a substantial excess of high $Lum OIII $ values. This result is also reflected in the analysis of the accretion strength onto central black holes. These findings indicate that external perturbations from a nearby galaxy companion can influence gas flows induced by galactic bars, leading to increased nuclear activity in barred AGN galaxies within pair systems. Thus, the coexistence of both — bars and interactions — significantly amplifies central nuclear activity, thereby influencing the accretion processes onto massive black holes.