We study a continuous-time stochastic Stackelberg control problem in which a leader steers a system of strategic followers through two non-standard channels - the information structure and a transfer mechanism - rather than through the dynamics directly. The latent environment is a jump-diffusion; the leader commits to a Gaussian public-signaling channel whose belief consequences are tracked by a finite-dimensional projection filter (the exact filter being infinite-dimensional), together with a Groves transfer that aligns the followers' incentives. Under truthful disclosure, efficient behavior is a dominant-strategy best response, and the induced differential game admits saturated and bang-bang Nash feedback. We cast the leader's distributionally robust problem, over a relative-entropy ambiguity neighborhood, as a two-controller Isaacs equation; prove that incentive alignment collapses the bilevel Stackelberg problem to a single robust control problem with an exact first-order condition; and characterize the value function as the unique viscosity solution, with a verification theorem valid for the non-smooth bang-bang feedback and a semiconcavity result that renders the switching set Lebesgue-null. We instantiate the framework on resilient multi-area power-system coordination under extreme weather. Calibrated to the 2021 Winter Storm Uri, an Isaacs solve over ERCOT's near-islanded interconnection (a 0.82 GW tie, under 2% of peak) shows mutual aid removes about 8% of social cost, rising to roughly 30% under the FERC/DOE-recommended interregional transfer capability; a reserve-scheduling experiment shows that public disclosure lowers welfare cost by 37% under autarky and 48% under market coupling, and that information design and market coupling are complements under common (systemic) risk.

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