This work presents a simulation-driven design optimization framework to address the inherent thermal challenges in Gallium Oxide (Ga₂O₃) devices and enhance their electrical performance. Ga₂O₃, an ultra-wide bandgap (UWBG) semiconductor, holds significant promise for high-power applications due to its high breakdown electric field and large bandgap. However, its low thermal conductivity - relative to other UWBG materials like Silicon Carbide (SiC) and Gallium Nitride (GaN) - limits its ability to dissipate heat efficiently. This thermal limitation leads to hotspot formation, inducing thermal stress and performance degradation over time. Excessive self-heating can result in threshold voltage shifts, increased leakage currents, and reduced electron mobility, ultimately compromising device efficiency. In extreme scenarios, self-heating may trigger thermal runaway, causing irreversible device damage. To overcome these challenges, this study integrates parallel GPU-accelerated, high-fidelity electrical-thermal-mechanical simulations within an adjoint-based optimization framework. The approach enables the prediction of device behavior under various operational conditions while optimizing for enhanced thermal management and electrical performance. Key optimization objectives include minimizing thermal-mechanical stresses along material interfaces, reducing localized heating, and improving overall device reliability. This framework offers a systematic pathway for advancing Ga₂O₃ device designs, ensuring greater operational stability and performance in high-power applications.