Dynamic correlation functions such as single particle Green's functions, linear response functions, or dynamical susceptibilities serve as a foundational tool kit for studying strongly correlated quantum systems. Ideally, a quantum computer will be able to extract these functions for systems sizes that are intractable on classical computers. When it comes to studying these functions at finite temperature the main bottleneck comes from the resource overhead and circuit complexities required in preparing each Gibbs sample. We present a framework aimed at alleviating this bottleneck by optimizing a series of approximations. Specifically, we sample from a series of time averaged embedded clusters initially in their respective local Gibbs states. After extracting each approximate dynamic correlation function we employ Richardson extrapolation where the error expanded in the series is determined by the total number of sub-clusters used in each approximation. We obtain higher order estimates for each distinct path of approximations. We can optimize even further by weighting each distinct path by how closely each path fits the proper fluctuation theorem. We demonstrate this toolbox numerically using exact diagonalization of the Hubbard model on small clusters.