Modern data centers (DCs) hosting artificial intelligence (AI)-dedicated devices operate at high power densities with rapidly varying workloads, making minute-level adaptation essential for safe and energy-efficient operation. However, manually designing piecewise deep reinforcement learning (DRL) agents cannot keep pace with frequent dynamics shifts and service-level agreement (SLA) changes of an evolving DC. This specification-to-policy lag causes a lack of timely, effective control policies, which may lead to service outages. To bridge the gap, we present DCoPilot, a hybrid framework for generative control policies in dynamic DC operation. DCoPilot synergizes two distinct generative paradigms, i.e., a large language model (LLM) that performs symbolic generation of structured reward forms, and a hypernetwork that conducts parametric generation of policy weights. DCoPilot operates through three coordinated phases: (i) simulation scale-up, which stress-tests reward candidates across diverse simulation-ready (SimReady) scenes; (ii) meta policy distillation, where a hypernetwork is trained to output policy weights conditioned on SLA and scene embeddings; and (iii) online adaptation, enabling zero-shot policy generation in response to updated specifications. Evaluated across five control task families spanning diverse DC components, DCoPilot achieves near-zero constraint violations and outperforms all baselines across specification variations. Ablation studies validate the effectiveness of LLM-based unified reward generation in enabling stable hypernetwork convergence.