Activated coolant in light water reactors generates significant radiation fields impacting occupational safety and plant design. This study presents a high-fidelity, dynamic model quantifying the activated coolant source term in the 330 MWth System-integrated Modular Advanced Reactor (SMART), an integral small modular reactor (SMR). A comprehensive isotopic inventory of key activation products 16N, 17N, 19O, 14C, 3H, 24Na, 38Cl, and 32P is established under normal operating conditions. Analysis reveals that 16N dominates short-lived activity, reaching 7.59×1013 Bq in the primary coolant, accounting for 96.8% of nitrogen-based activation. Tritium (3H) production exhibits pronounced operational mode dependence on chemical shim operation, yielding 3.49×1012 Bq, which is 18% higher than mechanical shim operation (2.96×1012 Bq) after a complete fuel cycle. Fission process contributes approximately 85% to the total 3H inventory. A comprehensive static and dynamic sensitivity analysis, perturbing parameters by 1.0%, identifies coolant flow rate, removal efficiency, and reactor power as the most influential parameters governing 3H behavior, with sensitivity coefficients of -0.969, -0.969, and +0.855, respectively. Dynamic sensitivity reveals that parameters related to 10B exhibit 72-88% temporal variability, necessitating explicit time-dependent treatment. Novel Shannon entropy analysis of 14 tritium-bearing state variables quantifies pathway-specific complexity. Chemical shim variables maintain constant dynamic entropy (11.34 bits), mechanical shim variables show monotonically increasing entropy (11.05 → 11.50 bits), and fission-to-gap pathways exhibit the highest variability (10.70 → 11.50 bits). Total system entropy stabilizes at 158.66 ± 0.31 bits within 10 days, demonstrating rapid equilibrium despite 3H pathway complexity.