Multi-UAV systems performing the task of transporting material with cable-suspended payloads in complex environments are susceptible to multi-source risk factors such as external disturbances and obstacles. To solve this problem, this paper proposes a risk-triggered distributed safety cooperative control method. Firstly, the disturbance caused by cable-suspended payloads is modeled and characterized, and a disturbance observer is designed to realize accurate estimation. Subsequently, considering comprehensively the stochastic uncertainties including position tracking, state estimation and obstacle perception, the coupled collision risk is quantitatively calculated based on Conditional Value-at-Risk (CVaR). Finally, a safety threshold is designed to evaluate the collision risk. Combined with robust formation control, collision avoidance strategy and disturbance compensation mechanism, a risk-triggered distributed safe cooperative controller is constructed. The effectiveness of the proposed method is verified through flight experiments and simulation tests. The results show that the proposed method significantly improves the control accuracy of multi-UAV systems under cable-suspended payload disturbances and the safety in complex environments, while maintaining low conservatism of obstacle avoidance.