We consider a capacitated location-allocation-pricing problem in a single-commodity supply chain with stochastic price-sensitive demands, where the location, allocation and pricing decisions are made simultaneously. Under a general risk measure representing an arbitrary risk tolerance policy, the problem is modeled as a two-stage stochastic mixed-integer program with a translation-invariant monotone risk measure. To solve the problem, we develop a risk-aware logic-based Benders decomposition (LBBD) method and demonstrate its applicability on three common risk measures for moderate, cautious, and pessimistic policies. To enhance the performance of the solution method, we introduce strengthened LBBD cuts and two families of problem-specific valid inequalities. Through comprehensive computational experiments, we demonstrate that our decomposition framework efficiently solves instances adapted from the literature, with algorithmic enhancements significantly improving solution times and scalability. The results reveal substantial advantages of our stochastic approach over traditional risk-measured methods, particularly for larger instances. The generality of our framework for incorporating risk measures makes it readily adaptable to other discrete optimization problems with similar structure.