We show that recent high-precision measurements of relative on-site interaction energies ΔU in a Mott insulator require a theoretical description beyond the standard Hubbard-model interpretation, when combined with an accurate coupled-channels calculation. In contrast to more sophisticated lattice models, which can be elaborate especially for parameter optimization searches, we introduce an easy to use effective description of U valid over a wide range of interaction strengths modeling atomic pairs confined to single lattice sites. This concise model allows for a straightforward combination with a coupled-channels analysis. With this model we perform such a coupled-channels analysis of high-precision Li7 spectroscopic data on the on-site interaction energy U, which spans over four Feshbach resonances and provide an accurate and consistent determination of the associated resonance positions. Earlier experiments on three of the Feshbach resonances are consistent with this analysis. Moreover, we verify our model with a more rigorous numerical treatment of the two atom system in an optical lattice.