Understanding various fundamental properties of nucleons and nuclei are among the most important scientific goals at the upcoming Electron-Ion Collider (EIC). With the unprecedented opportunity provided by the next-generation machine, the EIC might provide definitive answers to many standing puzzles and open questions in modern nuclear physics. Here we investigate one of the golden measurements proposed at the EIC, which is to obtain the spatial gluon density distribution within a lead ($Pb$) nucleus. The proposed experimental process is the exclusive $J/psi$ vector-meson production off the $Pb$ nucleus - $e+Pbrightarrow e+J/psi+Pb$. The Fourier transformation of the momentum transfer $|t|$ distribution of the coherent diffraction is the transverse gluon spatial distribution. In order to measure it, the experiment has to overcome an overwhelmingly large background arising from the incoherent diffractive production, where the nucleus $Pb$ mostly breaks up into fragments of particles in the far-forward direction close to the hadron-going beam rapidity. In this paper, we systematically study the rejection of incoherent $J/psi$ production by vetoing products from these nuclear breakups - protons, neutrons, and photons, which is based on the BeAGLE event generator and the most up-to-date EIC Far-forward Interaction Region design. The achieved vetoing efficiency, the ratio between the number of vetoed events and total incoherent events, ranges from about 80% - 99% depending on $|t|$, which can resolve at least the first minimum of the coherent diffractive distribution based on the Sar$it{t}$re model. Experimental and accelerator machine challenges as well as potential improvements are discussed.
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