We present comprehensive modeling of a SiGeSn multi-quantum well laser that has been previously experimentally shown to feature an order of magnitude reduction in the optical pump threshold compared to bulk lasers. We combine experimental material data obtained over the last few years with $k·p$ theory to adapt transport, optical gain, and optical loss models to this material system (drift-diffusion, thermionic emission, gain calculations, free carrier absorption, and intervalence band absorption). Good consistency is obtained with experimental data, and the main mechanisms limiting the laser performance are discussed. In particular, modeling results indicate a low non-radiative lifetime, in the 100 ps range for the investigated material stack, and lower than expected Γ-L energy separation and/or carrier confinement to play a dominant role in the device properties. Moreover, they further indicate that this laser emits in transverse magnetic polarization at higher temperatures due to lower intervalence band absorption losses. To the best of our knowledge, this is the first comprehensive modeling of experimentally realized SiGeSn lasers, taking the wealth of experimental material data accumulated over the past years into account. The methods described in this paper pave the way to predictive modeling of new (Si)GeSn laser device concepts.