Numerous experimental studies have found the presence of (Cu)-Ni-Mn-Si clusters in neutron irradiated reactor pressure vessel steels, prompting concerns that these clusters could lead to larger than expected increases in hardening, especially at high fluences late in life. The mechanics governing clustering for the Fe-Mn-Ni-Si system are not well-known; state-of-the-art methods use kinetic Monte Carlo (KMC) parameterized by density functional theory (DFT) and thermodynamic data to model the time evolution of clusters. However, DFT-based KMC studies have so far been limited to only pairwise interactions due to lack of DFT data. Here, we explicitly calculate the binding energy of triplet clusters of Mn, Ni, Cu, Si, and vacancies in bcc Fe using DFT to show that the presence of vacancies, Si, or Cu stabilizes cluster formation, as clusters containing exclusively Mn and/or Ni are not energetically stable in the absence of interstitials. We further identify which clusters may be reasonably approximated as a sum of pairwise interactions and which instead require an explicit treatment of the three-body interaction, showing that the three-body term can account for as much as 0.3 eV, especially for clusters containing vacancies.