Poly(2,2′-biquinoline-6,6′-dicarbohydrazide)-co-(bistrimelliteimide)methylene-bisanthranylide (PHI) and its metal–polymer complex PHI-Сu(I) containing several types of functional groups (hydrazide, carboxyl, amide, and imide fragments) were synthesized to prepare two types of dense nonporous membranes. The study on morphology using scanning electron microscopy (SEM), measurements of mechanical, thermal, and transport properties of the membrane samples was carried out. The main mechanical properties of both membranes do not differ significantly, but the values of ultimate deformation differ palpably as a result of a non-uniform character of the deformation process for the PHI membrane. The thermal analysis based on the curves of thermogravimetric (TGA) and differential thermal (DTA) analyses of the PHI and PHI-Cu(I) membranes revealed peculiarities of the membrane structure. Transport properties were studied in pervaporation (PV) of methanol (MeOH) and dimethyl carbonate (DMC) mixtures including an azeotropic point. Intrinsic properties of the penetrant–membrane system were also determined. It was found that the total flux is higher through the PHI membrane, but the PHI-Сu(I) membrane exhibits a higher separation factor. Calculation of the pervaporation separation index (PSI) allowed to conclude that the PHI-Сu(I) membrane exhibits better transport properties as compared with the PHI membrane.