The development of intermediate-temperature solid oxide fuel cells (IT-SOFCs) is currently a focus of research, as it has the potential to improve their operational characteristics, lower their cost, and enhance their stability. However, achieving high-performance functional materials at working temperatures of 600–800 °C poses a significant challenge. Barium cerate (BaCeO3) has been identified as a promising material for the solid electrolyte of IT-SOFCs due to its high proton conductivity among A2+B4+O3-type perovskites. Nevertheless, producing a single-phase and dense ceramic based on BaCeO3 with high conductivity in the intermediate-temperature range remains a challenge. This study employed the hydrazine-nitrate combustion method to obtain BaCeO3 precursor powder, which was then used to produce dense proton-conducting ceramics by isostatic pressing and subsequent isothermal sintering at temperatures of 1100–1500 °C. The resulting samples were characterized using various techniques, including XRD, SEM, Vickers microhardness, TMA, LFA, and impedance spectroscopy. The findings indicate that the proposed hydrazine-nitrate approach was successful in producing ceramic samples with an average crystallite size of 95–220 nm, an average grain size of 4.8–9.9 μm, and a relative density of 70.7–91.2% depending on the sintering conditions. The most densely sintered sample (91.2%) exhibited the highest mechanical and functional properties, with Vickers microhardness values of 3.0 GPa (R T.), a coefficient of thermal expansion of 8.7 × 10–6 (R T.-1000 °C), thermal conductivity of 1.51–1.06 W/m·°C (R.T.-1000 °C), and logarithmic conductivity of −3.37 to −6.35 S/cm (675–980 °C). Thus, the proposed hydrazine-nitrate approach shows promise for producing high-performance conducting ceramic materials for IT-SOFCs.