In the landscape of low-carbon transition in the power system, it is imperative for the system operator (SO) to implement electricity-carbon joint management. Currently, the increasing integration of renewable energy sources (RESs) is facilitating the achievement of emission reduction targets. However, the inherent uncertainties of RES power output pose challenges on power system operation, highlighting the needs for developing a power prediction model that serves as the prerequisite of better scheduling decisions. Nevertheless, existing accuracy-oriented prediction may not necessarily guarantee better decisions. Besides, due to the inevitable prediction errors, SOs have to adjust power outputs of thermal generators (TGs) during the intraday redispatching, leading to unexpected emission variations for each generation companies (GENCOs). Under the current centralized emission trading scheme (ETS), GENCOs with lower emissions are unable to fully utilize their emission allowances, while those exceeding their limits may face high penalties. However, these two groups of GENCOs exhibit inherent complementarity in terms of emission allowance consumption. To address the above challenges, this study proposes a novel multi-stage electricity-carbon joint management framework, where the power prediction model is decision-oriented to focus more on cost-saving. Moreover, bilateral trading contracts for emission allowances among GENCOs are incorporated into the proposed framework to promote the sufficient utilization of allocated emission allowances and prevent emission exceedances, thereby enhancing the total social welfare. Extensive simulations on a modified IEEE 30-bus system statistically verify the effectiveness of the developed decision-oriented predict-then-optimize method in terms of reducing operation cost. The welfare improvement of GENCOs brought by the designed bilateral trading contracts is also verified through simulation studies.