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In the Penrose process and the Blandford-Znajek mechanism, the rotational energy of a black hole (BH) is extracted via particle fission and magnetic tension, respectively. Recently, inspired by a fundamental trait in plasma astrophysics known as magnetic reconnection (MR), a new energy extraction mechanism based on the fast reconnection of the magnetic field lines inside the ergosphere has been proposed by Comisso and Asenjo. In this paper, we investigate energy extraction caused by MR in the ergosphere of a rapidly spinning BH with broken Lorentz symmetry by a background bumblebee vector field. The desired rotating BH solution differentiates from the standard Kerr BH via the Lorentz symmetry breaking (LSB) parameter $l$, which comes from nonminimal coupling between the bumblebee field with nonzero vacuum expectation value and gravity. We find that incorporating $l<0$ in the background is in the interest of the energy extraction via MR for the fast-spinning BH surrounded by the plasma with weak magnetization, below what is expected from the scenario by Comisso and Asenjo. Our analysis robustly indicates that the power of energy extraction and efficiency of the plasma energization process through fast MR is more efficient than the Comisso-Asenjo solution, provided that the LSB parameter is negative, $l<0$. Compared to the Blandford-Znajek mechanism arising from the underlying background, we also show that MR is a more efficient energy extraction mechanism if $l<0$.