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Eclipsing, spectroscopic double-lined (SB2) binaries remain to be the prime source of precise and accurate fundamental properties of stars. Furthermore, high-cadence spectroscopic observations of the eclipse phases allow us to resolve the Rossiter-McLaughlin effect whose modelling offers the means to probe spin-orbit misalignment in binaries. In this study, we develop the LSDBinary algorithm that is capable of working with both in-eclipse and out-of-eclipse spectra of SB2 binaries as input and delivers the LSD profiles, LSD-based model spectra, and precise RVs of both binary components as output. We offer an option to account for the Rossiter-McLaughlin effect in the calculation of the initial guess LSD profiles and components' flux ratio such that the effect can be modelled within the algorithm itself. We provide an extensive test of the LSDBinary software package on simulated spectra of artificial binaries. We study the effects of signal-to-noise-ratio of input spectra, resolving power of the instrument, uncertain atmospheric parameters of stars, and orbital properties of the binary system on the resulting LSD profiles and RVs measured from them. We find that atmospheric parameters have negligible effect on the shape of the computed LSD profiles while affecting mostly their global scaling. Our results are barely sensitive to signal-to-noise ratio of the input spectra provided they contain sufficient number of spectral lines, such as in A-type stars and later. Finally, the orbital inclination angle and components' radii ratio are found to have the largest effect on the shapes of the LSD profiles and RV curves extracted from them. The LSDBinary algorithm is specifically developed to perform detailed spectroscopic studies of eclipsing SB2 systems whose orbital configuration and components' atmospheric parameters are estimated by other means.