Tracking financial investments in climate adaptation is a complex and expertise-intensive task, particularly for Early Warning Systems (EWS), which lack standardized financial reporting across multilateral development banks (MDBs) and funds. To address this challenge, we introduce an LLM-based agentic AI system that integrates contextual retrieval, fine-tuning, and multi-step reasoning to extract relevant financial data, classify investments, and ensure compliance with funding guidelines. Our study focuses on a real-world application: tracking EWS investments in the Climate Risk and Early Warning Systems (CREWS) Fund. We analyze 25 MDB project documents and evaluate multiple AI-driven classification methods, including zero-shot and few-shot learning, fine-tuned transformer-based classifiers, chain-of-thought (CoT) prompting, and an agent-based retrieval-augmented generation (RAG) approach. Our results show that the agent-based RAG approach significantly outperforms other methods, achieving 87\% accuracy, 89\% precision, and 83\% recall. Additionally, we contribute a benchmark dataset and expert-annotated corpus, providing a valuable resource for future research in AI-driven financial tracking and climate finance transparency.