Millimeter-wave (mmWave) communication depends on highly directional beamforming, while fast mobility, blockage, and rapid geometry changes in vehicle-to-everything (V2X) scenarios make beam tracking challenging. In cooperative multi-base-station (BS) systems, conventional hierarchical methods usually separate BS selection and beam selection, which may cause error propagation when beam states change abruptly. To address this issue, this paper proposes Cooperative Radio Sensing with Large Language Models (CRS-LLM), a cooperative beam prediction framework for next-step joint BS-beam prediction. CRS-LLM formulates beam tracking as a single classification problem over the joint BS-beam space, avoiding cascaded decision errors. To adapt channel state information (CSI) to large language models, a dual-view CSI tokenizer extracts frequency-domain and delay-domain channel features through a lightweight CNN front-end and temporal tokenization module. A truncated GPT-style backbone is then used for temporal modeling with parameter-efficient adaptation. In addition, a transition-aware switch-gated predictor combines a stable branch, a residual flip branch, and a low-rank transition prior to capture both smooth evolution and abrupt changes. Simulation results show that CRS-LLM outperforms CSI-Transformer, Hierarchical BS-Beam, and representative CNN- and recurrent-neural-network baselines in Top-1 accuracy and normalized beam gain under different SNR conditions, while also showing strong few-shot performance and promising zero-shot transferability.