Accurate climate forecasting is vital for Bangladesh, a region highly susceptible to climate change impacts on temperature and rainfall. Existing models often struggle to capture long-range dependencies and complex temporal patterns in climate data. This study introduces an advanced Long Short-Term Memory (LSTM) model integrated with an attention mechanism to enhance the prediction of temperature and rainfall dynamics. Utilizing comprehensive datasets from 1901-2023, sourced from NASA's POWER Project for temperature and the Humanitarian Data Exchange for rainfall, the model effectively captures seasonal and long-term trends. It outperforms baseline models, including XGBoost, Simple LSTM, and GRU, achieving a test MSE of 0.2411 (normalized units), MAE of 0.3860 degrees C, R^2 of 0.9834, and NRMSE of 0.0370 for temperature, and MSE of 1283.67 mm^2, MAE of 22.91 mm, R^2 of 0.9639, and NRMSE of 0.0354 for rainfall on monthly forecasts. The model demonstrates improved robustness with only a 20 percent increase in MSE under simulated climate trends (compared to an approximately 2.2-fold increase in baseline models without trend features) and a 50 percent degradation under regional variations (compared to an approximately 4.8-fold increase in baseline models without enhancements). These results highlight the model's ability to improve forecasting precision and offer potential insights into the physical processes governing climate variability in Bangladesh, supporting applications in climate-sensitive sectors.