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Intense, quasi-monochromatic, polarized $γ$-ray beams produced by Compton scattering of laser photons against relativistic electrons are used for fundamental studies and applications. Following a series of photoneutron cross section measurements in the Giant Dipole Resonance energy region performed at the NewSUBARU synchrotron radiation facility, we have developed the eliLaBr Monte Carlo simulation code for characterization of the scattered $γ$-ray photon beams. The code is implemented using Geant4 and is available on the GitHub repository (https://github.com/dan-mihai-filipescu/eliLaBr). Here we report the validation of the eliLaBr code on NewSUBARU LCS $γ$-ray beam flux and spectral distribution data and two applications performed with it for asymmetric transverse emittance profiles electron beams, characteristic for synchrotrons. The first application is based on a systematic investigation of transverse collimator offsets relative to the laser and electron beam axis. We show that the maximum energy of the LCS $γ$-ray beam is altered by vertical collimator offsets, where the edge shifts towards lower energies with the increase in the offset. Secondly, using the eliLaBr code, we investigate the effect of the laser polarization plane orientation on the properties of the LCS $γ$-ray beams produced with asymmetric emittance electron beams. We show that: 1. The use of vertically polarized lasers contributes to the preservation of the LCS $γ$-ray beam maximum energy edge by increasing the precision in the vertical collimator alignment. 2. Under identical conditions for the electron and laser beams phase-space distributions, the energy spectrum of the scattered LCS $γ$-ray beam changes with the laser beam polarization plane orientation: the use of vertically polarized laser beams slightly deteriorates the LCS $γ$-ray beam energy resolution.