Cooperative driving is a safety- and efficiency-critical task that requires the coordination of diverse, interaction-realistic multi-agent trajectories. Although existing diffusion-based methods can capture multimodal behaviors from demonstrations, they often exhibit weak scene consistency and poor alignment with closed-loop cooperative objectives. This makes post-training necessary for further improvement, yet achieving stable online post-training in reactive multi-agent environments remains challenging. In this paper, we propose SCORP, a scene-consistent multi-agent diffusion planner with stable online reinforcement learning (RL) post-training for cooperative driving. For pre-training, we develop a scene-conditioned multi-agent denoising architecture that couples inter-agent self-attention with a dual-path conditioning mechanism: cross-attention provides direct scene-information injection, while AdaLN-Zero enables additional flexible and stable conditional modulation, thereby improving the scene consistency and road adherence of joint trajectories. For post-training, we formulate a two-layer Markov decision process (MDP) that explicitly integrates the reverse denoising chain with policy-environment interaction. We further co-design dense, well-shaped planning rewards and variance-gated group-relative policy optimization (VG-GRPO) to mitigate advantage collapse and gradient instability during closed-loop training. Extensive experiments show that SCORP outperforms strong open-source baselines on WOMD, with 10.47%-28.26% and 1.70%-7.22% improvements in core safety and efficiency metrics, respectively. Moreover, compared with alternative post-training methods, SCORP delivers significant and consistent gains in both driving safety and traffic efficiency, highlighting stable and sustained advances in closed-loop cooperative driving.