Exploration of the lunar south pole with a solar-powered rover is challenging due to the highly dynamic solar illumination conditions and the presence of permanently shadowed regions (PSRs). In turn, careful planning in space and time is essential. Mission-level path planning is a global, spatiotemporal paradigm that addresses this challenge, taking into account rover resources and mission requirements. However, existing approaches do not proactively account for random disturbances, such as recurring faults, that may temporarily delay rover traverse progress. In this paper, we formulate a chance-constrained mission-level planning problem for the exploration of PSRs by a solar-powered rover affected by random faults. The objective is to find a policy that visits as many waypoints of scientific interest as possible while respecting an upper bound on the probability of mission failure. Our approach assumes that faults occur randomly, but at a known, constant average rate. Each fault is resolved within a fixed time, simulating the recovery period of an autonomous system or the time required for a team of human operators to intervene. Unlike solutions based upon dynamic programming alone, our method breaks the chance-constrained optimization problem into smaller offline and online subtasks to make the problem computationally tractable. Specifically, our solution combines existing mission-level path planning techniques with a stochastic reachability analysis component. We find mission plans that remain within reach of safety throughout large state spaces. To empirically validate our algorithm, we simulate mission scenarios using orbital terrain and illumination maps of Cabeus Crater. Results from simulations of multi-day, long-range drives in the LCROSS impact region are also presented.