The presented research was focused on the development of a new method of sandwich structure manufacturing involving FDM-printing (fused deposition modeling) techniques and compression molding. The presented concept allows for the preparation of thermoplastic-based composites with enhanced mechanical properties. The sample preparation process consists of 3D printing the sandwich’s core structure using the FDM method. For comparison purposes, we used two types of GPET (copolymer of polyethylene terephthalate)-based filaments, pure resin, and carbon fiber (CF)-reinforced filaments. The outer reinforcing layer “skins” of the sandwich structure were prepared from the compression molded prepregs made from the LCP (liquid-crystal polymer)-fiber fabric with the GPET-based matrix. The final product consisting of an FDM-printed core and LCP-based prepreg was prepared using the compression molding method. The prepared samples were subjected to detailed materials analyses, including thermal analyses (thermogravimetry-TGA, differencial scanning calorimetry-DSC, and dynamic thermal-mechanical analysis-DMTA) and mechanical tests (tensile, flexural, and impact). As indicated by the static test results, the modulus and strength of the prepared composites were slightly improved; however, the stiffness of the prepared materials was more related to the presence of the CF-reinforced filament than the presence of the composite prepreg. The main advantage of using the developed method is revealed during impact tests. Due to the presence of long LCP fibers, the prepared sandwich samples are characterized by very high impact resistance. The impact strength increased from 1.7 kJ/m2 for pure GPET samples to 50.4 kJ/m2 for sandwich composites. For GPET/CF samples, the increase is even greater. The advantages of the developed solution were illustrated during puncture tests in which none of the sandwich samples were pierced.