The atomic beam probe (ABP) is a beam diagnostic concept that opens opportunities in plasma edge measurements due to the sensitivity of the magnetic field and the high temporal resolution. The first ABP has been installed and is operating on the COMPASS tokamak. A new numerical toolset, which is the subject of this paper, was required to model the diagnostic to accurately detect the alkali beam. For further development and understanding of the diagnostic, this tool had to be designed to simulate different magnetic field configurations in a performance-efficient manner. The TAIGA synthetic diagnostic (TAIGA-SD), which was implemented with a massively parallel trajectory solver core that runs on graphic cards to support experiments, provides a better understanding of measurements and has opened opportunities for future applications. This paper presents the model concept with relevant physical processes and necessary simplifications. The submodules implemented or integrated into the synthetic diagnostic are explained and described, and their scopes of validity are highlighted. This includes the integration of RENATE-OD for the primary ionization radial distribution for lithium beams, as well as the implementation and verification of a combined electron impact and charge exchange ionization module for other alkaline beams, which is a new atomic physics solver. Calculations were performed to investigate the relation between magnetic field, electron density, and temperature perturbations. Further simulations were run to estimate beam attenuation due to secondary ionization. The utilization of the ABP synthetic diagnostic is demonstrated by comparing it with the measurements.