The automatic classification of sleep stages and Cyclic Alternating Pattern (CAP) subtypes from electroencephalogram (EEG) recordings remains a significant challenge in computational sleep research because of the short duration of CAP events and the inherent class imbalance in clinical datasets. Background/Objectives: The research introduces a domain-specific deep learning system that employs an LSTM network optimized through a PSO-Hyperband hybrid hyperparameter tuning method. Methods: The research enhances EEG-based sleep analysis through the implementation of hybrid optimization methods within an LSTM architecture that addresses CAP sequence classification requirements without requiring architectural changes. Results: The developed model demonstrates strong performance on the CAP Sleep Database by achieving 97% accuracy for REM and 96% accuracy for stage S0 and ROC AUC scores exceeding 0.92 across challenging CAP subtypes (A1–A3). The model transparency is improved through the application of SHAP-based interpretability techniques, which highlight the role of spectral and morphological EEG features in classification outcomes. Conclusions: The proposed framework demonstrates resistance to class imbalance and better discrimination between visually similar CAP subtypes. The results demonstrate how hybrid optimization methods improve the performance, generalizability, and interpretability of deep learning models for EEG-based sleep microstructure analysis.