Supervised fine-tuning (SFT) and reinforcement learning with verifiable rewards (RLVR) are two widely used post-training paradigms for improving the reasoning ability of large language models (LLMs). Recent methods attempt to integrate SFT and RLVR in a single stage by reweighting or scheduling their objectives. However, such coupling can be counterproductive because supervised updates are not uniformly beneficial for reward optimization. To address this, we propose BRIDGE, a scalable framework in which SFT learns to supervise RL by selectively transferring knowledge that improves reward optimization. Specifically, BRIDGE alternates two updates at each meta-training step: a base-model update that fuses the SFT and RL gradients, and an update to a lightweight low-rank adapter (LoRA) that coordinates the two objectives by maximizing a cooperative-gain signal, defined as the reward of joint SFT-RL training over an RL-only baseline. Across five mathematical reasoning benchmarks, BRIDGE consistently outperforms two-stage cold start, naive mixing, and representative single-stage integration baselines, yielding over three points average absolute improvement and more stable training dynamics. We further show that BRIDGE extends to logical reasoning and generalizes out-of-distribution to code and science without additional training, while staying robust under noisy rewards.