The reversible operation of a commercialized molten carbonate fuel cell as an electrolyzer is attractive to store the surplus of renewable energy and convert CO2 into valuable fuel, like syngas (H2 and CO). This work aims to assess this capability by developing a model for the co-electrolysis of CO2 and H2O in the molten carbonate electrolysis cell (MCEC). Furthermore, a 1 MW power-to-gas system is simulated and optimized to maximize the process efficiency via a profound sensitivity analysis and a pinch heat integration analysis. The study's results reveal that when CO2 electrolysis occurs, there is a risk of reaching the limiting current density at lower values, thus affecting the cell's performance. As for the simulation, the process yields an efficiency of 68.87% with an electrolyzer's power density of 0.082 W/cm2 per cell, producing 1.3 tons/day of methane gas ready for injection into the natural gas network.