The Dempster-Shafer theory is a well-known mathematical framework for modeling situations involving partially specified information, that generalizes classical probability. The theory of probability boxes (p-boxes) is a distinguished part of this theory, since natural extensions of p-boxes reveal to be special belief functions. We consider the problem of approximating an arbitrary belief function with a “closest” natural extension under some constraints. The resulting approximation seeks to preserve the same information of the p-box induced by the initial belief function and to satisfy given upper bounds on the corresponding lower and upper Value-at-Risk (VaR) risk measures, defined as generalized inverse functions. The quoted approximation problem can be faced through a generalization of the classical optimal transport problem and the related Wasserstein distance. Then, the computation of the approximating p-box can be carried out efficiently through a generalization of the Dykstra’s algorithm by relying on a proper entropic formulation. We apply the described approximation on an ambiguous stop-loss reinsurance problem modeled as a Stackelberg game between a reinsurer, who acts as leader, and an insurer, who acts as follower.