Efficient channel state information (CSI) compression is essential in frequency division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems due to the substantial feedback overhead. Recently, deep learning-based compression techniques have demonstrated superior performance for CSI feedback. However, their performance often degrades under distribution shifts across wireless environments, largely due to limited generalization capability. To address this challenge, we consider a full-model fine-tuning scheme, in which both the encoder and decoder are jointly updated using a small number of recent CSI samples from the target environment. A key challenge in this setting is the transmission of updated decoder parameters to the receiver, which introduces additional communication overhead. To mitigate this bottleneck, we explicitly incorporate the bit rate of model updates into the fine-tuning objective and entropy-code the model updates jointly with the compressed CSI. Furthermore, we employ a structured prior that promotes sparse and selective parameter updates, thereby significantly reducing the model-update communication cost. Simulation results across multiple CSI datasets demonstrate that full-model fine-tuning substantially improves the rate-distortion performance of neural CSI compression, despite the additional cost of model updates. We further analyze the impact of the evaluation horizon, the quantization resolution of model updates, and the size of the target-domain dataset on the overall feedback efficiency.