Carbon dioxide-based geothermal systems use captured carbon dioxide as a subsurface working fluid, coupling renewable energy production with long-term carbon dioxide utilization. Their decades-long sustainability hinges on how carbon dioxide injection rate is allocated across wells and operational stages, yet each candidate strategy demands a computationally prohibitive full-physics simulation. Using the lifetime average reservoir exergy rate as the objective, Bayesian optimization is employed to make this search tractable, identifying interpretable interwell injection-rate allocations while matching the solution quality of an industry-standard optimizer at a lower computational cost. The framework is demonstrated on a geothermal reservoir of five injectors and two producers over a 30-year lifetime, comparing Uniform, Zonal, and Stepwise Injection Rate Schemes against the lifetime average reservoir exergy rate. The results demonstrate that the exergy optimization primarily navigates a fundamental trade-off between the quantity and the quality of the produced energy. Excessive injection greatly penalizes performance, reducing the lifetime average exergy rate to 429 kW at a tenfold-higher rate, below the 494 kW obtained under conservative injection, which nonetheless underutilizes the reservoir. Temporal flexibility in the injection-rate allocation resolves this tension. Relative to the Zonal Rate Scheme, the Stepwise Rate Scheme raises specific exergy by 15.6% and cuts carbon dioxide injection by 11% while still delivering a higher average exergy rate, thereby extracting more useful energy from less carbon dioxide circulation. Flow diagnostics and Proper Orthogonal Decomposition analyses further confirm the underlying mechanism. Guided by the well connectivity, optimized injection-rate allocations redirect injection spatially and temporally, extending fluid residence time, whereas the geologically anchored dominant mode of the flow field remains unchanged. This study contributes essential insights into advancing sustainable energy production techniques, and the framework is generalizable to other subsurface systems involving multiple injection and production wells.