Accurately forecasting carbon prices is essential for informed energy market decision-making, guiding sustainable energy planning, and supporting effective decarbonization strategies. However, it remains challenging due to structural breaks and high-frequency noise caused by frequent policy interventions and market shocks. Existing studies, including the most recent baseline approaches, have attempted to incorporate breakpoints but often treat denoising and modeling as separate processes and lack systematic evaluation across advanced deep learning architectures, limiting the robustness and the generalization capability. To address these gaps, this paper proposes a comprehensive hybrid framework that integrates structural break detection (Bai-Perron, ICSS, and PELT algorithms), wavelet signal denoising, and three state-of-the-art deep learning models (LSTM, GRU, and TCN). Using European Union Allowance (EUA) spot prices from 2007 to 2024 and exogenous features such as energy prices and policy indicators, the framework constructs univariate and multivariate datasets for comparative evaluation. Experimental results demonstrate that our proposed PELT-WT-TCN achieves the highest prediction accuracy, reducing forecasting errors by 22.35% in RMSE and 18.63% in MAE compared to the state-of-the-art baseline model (Breakpoints with Wavelet and LSTM), and by 70.55% in RMSE and 74.42% in MAE compared to the original LSTM without decomposition from the same baseline study. These findings underscore the value of integrating structural awareness and multiscale decomposition into deep learning architectures to enhance accuracy and interpretability in carbon price forecasting and other nonstationary financial time series.