Service-level requirements play a crucial role in eliminating stock-outs in a production pipeline. However, delivering a specific service level can become an unattainable goal given the various uncertainties influencing both the production pipeline and customer demand and causing the manufacturer to adapt the initial strategy in response to disruptions. Such deviations from optimality frequently result in unexpected (and potentially very high) costs and are complex to manage. On the one hand, the manufacturer can use a robust or distributionally robust approach to prepare for the worst-case disruption, ensuring that the realized cost will be lower than the estimated cost with high probability. On the other hand, this solution may lead to overly conservative production schedules. In this article, we take a different approach and develop a bilevel stochastic optimization model with chance constraints, which allows us to make the production more predictable in the event of disruptions by driving costs and optimal schedules closer to the benchmark for each scenario considered. We introduce doubly probabilistic service-level requirements to account for two interdependent layers of uncertainty, that is, production disruptions and distributional uncertainties in customer demand. This allows us to make high-quality production decisions with only a limited understanding of the demand pattern. Approximating the problem for a numerical solution, we guarantee tight optimality gaps for high service levels and propose an efficient solution scheme, combining robust scenario reduction with a customized Benders decomposition procedure. In the managerial section, we use the Omegax Swatch MoonSwatches example to demonstrate that a desirable doubly probabilistic service level can be attained for disruptions with a drop in demand. In the case of disruptions followed by a peak in demand, one can tighten the optimality gaps if the service level is reduced.