Reliable and interpretable tumor classification from clinical imaging remains a core challenge. The main difficulties arise from heterogeneous modality quality, limited annotations, and the absence of structured anatomical guidance. We present REACT-KD, a Region-Aware Cross-modal Topological Knowledge Distillation framework that transfers supervision from high-fidelity multi-modal sources into a lightweight CT-based student model. The framework employs a dual teacher design. One branch captures structure-function relationships through dual-tracer PET/CT, while the other models dose-aware features using synthetically degraded low-dose CT. These branches jointly guide the student model through two complementary objectives. The first achieves semantic alignment through logits distillation, and the second models anatomical topology through region graph distillation. A shared CBAM3D module ensures consistent attention across modalities. To improve reliability in deployment, REACT-KD introduces modality dropout during training, which enables robust inference under partial or noisy inputs. As a case study, we applied REACT-KD to hepatocellular carcinoma staging. The framework achieved an average AUC of 93.5\% on an internal PET/CT cohort and maintained 76.6\% to 81.5\% AUC across varying levels of dose degradation in external CT testing. Decision curve analysis further shows that REACT-KD consistently provides the highest net clinical benefit across all thresholds, confirming its value in real-world diagnostic practice. Code is available at: https://github.com/Kinetics-JOJO/REACT-KD