Accurately describing the relative energetics of alternative bis(μ-oxo) and μ-η2:η2 peroxo isomers of Cu2O2 cores supported by 0, 2, 4, and 6 ammonia ligands is remarkably challenging for a wide variety of theoretical models, primarily owing to the difficulty of maintaining a balanced description of rapidly changing dynamical and nondynamical electron correlation effects and a varying degree of biradical character along the isomerization coordinate. The completely renormalized coupled-cluster level of theory including triple excitations and extremely efficient pure density functional levels of theory quantitatively agree with one another and also agree qualitatively with experimental results for Cu2O2 cores supported by analogous but larger ligands. Standard coupled-cluster methods, such as CCSD(T), are in most cases considerably less accurate and exhibit poor convergence in predicted relative energies. Hybrid density functionals significantly underestimate the stability of the bis(μ-oxo) form, with the magnitude of the error being directly proportional to the percentage Hartree−Fock exchange in the functional. Single-root CASPT2 multireference second-order perturbation theory, by contrast, significantly overestimates the stability of bis(μ-oxo) isomers. Implications of these results for modeling the mechanism of C−H bond activation by supported Cu2O2 cores, like that found in the active site of oxytyrosinase, are discussed.