Existing dynamic constrained multi-objective evolutionary algorithms (DCMOEAs) generally suffer from unclear identification of dynamic characteristics and inadequate adaptability of prediction strategies, which restrict their optimization performance. To address these issues, this paper proposes a dynamic constrained multi-objective optimization algorithm for Bayesian-assisted classification prediction (BACP-DCMOA). First, a classification-prediction strategy is designed to categorize environmental changes into objective-dominant, constraint-dominant, mixed, and static/weak dynamic changes. Based on this classification, a hierarchical prediction strategy is constructed, in which customized predictors are matched according to the type and severity of environmental changes to improve the accuracy of tracking the shifting constrained Pareto front. Finally, Bayesian optimization is introduced to automatically optimize the relevant parameters. To evaluate the algorithm’s performance, comparative experiments are conducted on two widely used benchmark test suites. Experimental results show that BACP-DCMOA outperforms existing state-of-the-art algorithms in terms of dynamic tracking accuracy, convergence speed, and performance stability, providing a more efficient optimization solution for complex dynamic constrained multi-objective optimization problems (DCMOPs).