Humanoid robots are increasingly expected to operate in real-world environments that demand heavy-duty whole-body transport, including logistics, construction, and industrial material handling. However, transporting large, heavy, and unwieldy objects fundamentally changes the dynamics of humanoid control. Strong robot-payload coupling can cause substantial shifts in the combined center of mass, amplify angular momentum, and induce destabilizing whole-body oscillations, posing major challenges for existing loco-manipulation controllers. We introduce Unwieldy-Bench, a benchmark for stress-testing humanoid transport under heavy, extended, and irregular payloads. Our analysis shows that representative imitation-based controllers degrade sharply in these settings, as conventional methods primarily optimize pose and trajectory tracking while largely overlooking the evolution of whole-body momentum. Motivated by this observation, we propose DDM, a lightweight dynamics-aware reward-shaping strategy that regularizes payload-induced angular momentum. Experiments demonstrate that DDM improves transport stability, reduces object-drop rates, and expands the feasible payload range across diverse unwieldy transportation scenarios.