Considering the distinct differences in intrinsic characteristics (e.g., energy efficiency, power density, and response time), the synergy operation of combined hydrogen (H2) and battery systems within the source-grid-load-storage framework offers a promising solution to stabilize intermittent renewable energy supply, mitigate grid power fluctuations, and enhance system independence. However, the idling power of the H2 system (fuel cell and electrolyzer) poses significant limitations on H2 charging/discharging from intermittent renewable power sources, thereby affecting overall system performance. However, the current literature provides little progress on how to effectively self-consume onsite renewable energy and supply the electricity demand with high energy/power density in a fast response. To address these challenges, this study proposes and applies the H2-battery compensation operation in the hybrid H2-battery energy storage system to mitigate the adverse effects of idling power during low-carbon transformations. Additionally, transient degradation models for the battery and fuel cell (FC) are integrated into the system to dynamically quantify real-time aging, thereby avoiding performance overestimation. Moreover, the study also includes a comparative analysis of the pre-estimation on the H2 tank pressure level, and a parametrical analysis of the HV number and battery capacity. Results show that, compared to the isolated H2 energy storage system, the H2-battery synergy operation reduces the grid demand shortage coverage ratio from 91.01% to 57.8%. Furthermore, the proposed H2-battery compensation operation effectively overcomes the limitations of idling power during power charging/discharging of the H2 system, resulting in an increase of H2 demand shortage coverage ratio from 8.9% to 17.29%, and the decrease in battery degradation ratio from 1.73% to 1.65%. Afterward, a novel indicator, degradation cost per kWh energy, is proposed to quantify the techno-economic performance of the combined battery and H2 system. According to the results, the techno-economic feasibility is dependent on the proportion of battery and H2 capacities, with the FC ranging from 0.49 to 1.15 CNY/kWh and the battery ranging from 0.82 to 0.91 CNY/kWh. The research results provide valuable frontier guidelines for the design and operation of multi-energy systems, as well as innovative energy-sharing strategies, to achieve sustainable and carbon–neutral transformation.