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We present the quasi-particle self-consistent GW plus Gutzwiller rotationally invariant slave-boson (QSGW+G) method as a simplified version of QSGW + dynamical mean-field theory (DMFT). The application of QSGW+G to the thermoelectric material FeSb₂ yields an electronic structure consistent with QSGW+DMFT, albeit with some difference in fine details. It further reveils that the indirect band gap size is much more sensitive to Hund's coupling J rather than Hubbard interaction U. Most importantly, QSGW+G generates the results in a tiny fraction of time needed by QSGW+DMFT, rendering it an efficient screening method before higher-level QSGW+DMFT calculations in search of real correlated functional materials.