SABRE (Sodium iodide with Active Background REjection) is a direct detection dark matter experiment based on arrays of radio-pure NaI(Tl) crystals. The experiment aims at achieving an ultra-low background rate and its primary goal is to confirm or refute the results from the DAMA/LIBRA experiment. The SABRE Proof-of-Principle phase was carried out in 2020-2021 at the Gran Sasso National Laboratory (LNGS), in Italy. The next phase consists of two full-scale experiments: SABRE South at the Stawell Underground Physics Laboratory, in Australia, and SABRE North at LNGS. This paper focuses on SABRE South and presents a detailed simulation of the detector, which is used to characterise the background for dark matter searches including DAMA/LIBRA-like modulation. We estimate an overall background of 0.72 cpd/kg/keV$_{ee}$ in the energy range 1$-$6 keV$_{ee}$ primarily due to radioactive contamination in the crystals. Given this level of background and considering that the SABRE South has a target mass of 50 kg, we expect to exclude (confirm) DAMA/LIBRA modulation at $4~(5)\sigma$ within 2.5 years of data taking.
