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The Brillouin instability (BI) due to stimulated Brillouin scattering (SBS) and the transverse (thermal) mode instability (TMI) due to stimulated thermal Rayleigh scattering (STRS) limit the achievable power in high-power lasers and amplifiers. The pump power threshold for BI increases as the core diameter increases, but the threshold for TMI may decrease as the core diameter increases. In this paper, we use a multi-time-scale approach to simultaneously model BI and TMI, which gives us the ability to find the fiber diameter with the highest power threshold. We formulate the equations to compare the thresholds of the combined and individual TMI and BI models. At the pump power threshold and below, there is a negligible difference between the full and individual models, as BI and TMI are not strong enough to interact with each other. The highest pump threshold occurs at the optimal core size of 43µm for the simple double-clad geometry that we considered. We found that both effects contribute equally to the threshold, and the full BI and TMI model yields a similar threshold as the BI or TMI model alone. However, once the reflectivity is sufficiently large, we find in the full BI and TMI model that BI may trigger TMI and reduce the TMI threshold to a value lower than is predicted in simulations with TMI alone. This result cannot be predicted by models that consider BI and TMI separately. Our approach can be extended to more complex geometries and used for their optimization.