The information era imposes increasing demands on speed and energy efficiency, pushing conventional electronics to its physical limits. Silicon photonics offers a promising path forward, particularly when integrated with electro-optic ferroelectric materials, which can overcome fundamental bottlenecks in data rate and power consumption. With pronounced electro-optic activity and controllable ferroelectric domain orientation, thin-film barium titanate (BTO) presents a promising platform for advanced photonic integration. Here, we fabricate epitaxial BTO films on MgO substrates and use precise structural design to realize silicon nitride-BTO (SiN-BTO) hybrid microring devices. Through co-optimization of material synthesis and device architecture, we achieve a record-low power consumption of 0.0015 nW/pm─among the best reported values for electro-optic tuners and exhibiting the highest effective electro-optic coefficient for BTO-on-MgO platforms. Furthermore, we demonstrate nonvolatile tuning enabled by ferroelectric domain control, achieving an eight-level photonic device stable for over 12 h and optically readable switching energy of 0.191 pJ. This work establishes a versatile platform integrating the multifunctionality of ferroelectric BTO with energy-efficient photonic operation, providing a foundation for scalable circuits in next-generation communication, sensing, and computing systems.