The present study pays attention to biogas dry reforming for the purpose of producing H2. It is known that biogas contains approximately 40 vol% CO2, causing a decrease in the efficiency of power generation due to its lower heating value compared to natural gas, i.e., CH4. We suggest a hybrid system composed of a biogas dry reforming membrane reactor and a high-temperature fuel cell, i.e., a solid oxide fuel cell (SOFC). Since biogas dry reforming is an endothermic reaction, we adopt a membrane reactor, controlled by providing a non-equilibrium state via H2 separation from the reaction site. The purpose of the present study is to understand the effect of the thickness of the Pd/Cu membrane on the performance of the biogas dry reforming membrane reactor with a Pd/Cu membrane as well as a Ni/Cr catalyst. The impact of the reaction temperature, the molar ratio of CH4:CO2 and the differential pressure between the reaction chamber and the sweep chamber on the performance of the biogas dry reforming membrane reactor with the Pd/Cu membrane as well as the Ni/Cr catalyst was investigated by changing the thickness of the Pd/Cu membrane. It was revealed that we can obtain the highest concentration of H2, of 122,711 ppmV, for CH4:CO2 = 1:1 at a reaction temperature of 600 °C and a differential pressure of 0 MPa and using a Pd/Cu membrane with a thickness of 40 μm. Under these conditions, it can be concluded that the differential pressure of 0 MPa provides benefits for practical applications, especially since no power for H2 separation is necessary. Therefore, the thermal efficiency is improved, and additional equipment, e.g., a pump, is not necessary for practical applications.