The cosmological lithium problem (CLP) stems from the outstanding discrepancy between theoretical predictions and astronomical observations of primordial lithium abundances. For the radiogenic production of 7Li, 7Be plays a pivotal role in the Big Bang nucleosynthesis (BBN). Nevertheless, the data for neutron-induced 7Be destruction processes were still sparse, and especially lacked information on the contributions of transitions to the 7Li excited states. In this work, we have determined the 7Be\({(n,{p}_{0})}^{7}\)Li, 7Be\({(n,{p}_{1})}^{7}\)Li*, and 7Be\({(n,\alpha )}^{4}\)He reaction cross sections by means of the Trojan Horse method. The present and the previous data were analyzed together by a multichannel R-matrix fit, providing an improved uncertainty evaluation of the \((n,{p}_{0})\) channel and the first-ever quantification of the \((n,{p}_{1})\) contribution in the BBN-relevant energy range. We implemented the revised total reaction rate summing both the \((n,{p}_{0})\) and \((n,{p}_{1})\) contributions in a state-of-the-art BBN code PRIMAT. As a consequence, the present nuclear-physics data offers a reduction of the predicted 7Li abundance by about one-tenth, which would impose a stricter constraint on BBN and head us in the correct direction to the CLP solution.