Deliverability is recognized as the primary productivity indicator of salt cavern gas storage. Accurate prediction of deliverability based on input parameters such as total gas storage capacity and base gas capacity is essential for ensuring operational safety and achieving precise injection–withdrawal regulation. To address existing limitations of the conventional random forest (RF) model in deliverability prediction, including long optimization time, incomplete feature engineering, and insufficient generalization capability, a Multi-Sampling-Rate Random Forest (MSR-RF) prediction model is proposed in this study. The model is designed under a “triple-optimization” technical framework that integrates a dual-engine hyperparameter optimization strategy, a physics-informed and data-driven feature engineering scheme, and a three-tier adaptive sampling ensemble mechanism. These enhancements collectively improve the predictive accuracy, interpretability, and robustness of the model. Validation is performed using 1,334 samples, and the results show that the proposed model achieves a coefficient of determination of 0.9997 on the test set, demonstrating high prediction accuracy. Compared with the Single-Sampling-Rate RF model, reductions of 48.4% in root mean square error and 52.1% in mean absolute error are observed, and stable performance is maintained across various noise levels. This study provides new insights into the application of machine learning for intelligent storage management and predictive control in salt cavern gas storage systems.