Coarse-grained (CG) simulations are an important computational tool in chemistry and materials science. Recently, systematic "bottom-up" CG models have been introduced to capture electronic structure variations of molecules and polymers at the CG resolution. However, the performance of these models is limited by the ability to select reduced representations that preserve electronic structure information, which remains a challenge. We propose two methods for (i) identifying important electronically coupled atomic degrees of freedom and (ii) scoring the efficacy of CG representations used in conjunction with CG electronic predictions. The first method is a physically motivated approach that incorporates nuclear vibrations and electronic structure derived from simple quantum chemical calculations. We complement this physically motivated approach with a machine learning technique based on the marginal contribution of nuclear degrees of freedom to electronic prediction accuracy using an equivariant graph neural network. By integrating these two approaches, we can both identify critical electronically coupled atomic coordinates and score the efficacy of arbitrary CG representations for making electronic predictions. We leverage this capability to make a connection between optimized CG representations and the future potential for "bottom-up" development of simplified model Hamiltonians incorporating nonlinear vibrational modes.