Energy functionals of the Green's function can provide simultaneously spectral and thermodynamic properties of interacting electrons' systems. In this work, exploiting a localized-GW approach, we introduce an approximation to the exchange-correlation part of the Klein functional that generalizes the energy functional of DFT+U to host a dynamical screened potential U(ω). Furthermore, we solve exactly the resulting Dyson equation extending to crystals the algorithmic-inversion method. To demonstrate the approach we compute the spectral, thermodynamic, and vibrational properties of SrVO₃, finding results in excellent agreement with experiments and state-of-the-art computational methods, and overall at a negligible computational cost.