Cooperative benefit allocation becomes difficult when coalition worths are imprecise and decision rights are asymmetric. This study develops a power-weighted interval-valued core framework that integrates asymmetric voting power with satisfaction-based interval coalition rationality. Individual voting power is derived from coalition criticality in a weighted voting game and aggregated at the coalition level through a least-squares rule, while the resulting max-min allocation model is solved using a bisection procedure based on monotonic feasibility. In a stylized, normalized three-player example motivated by marine fishery insurance, one optimal allocation generated by the proposed model satisfies the adopted efficiency and coalitional rationality conditions, whereas a selected power-adjusted interval Shapley benchmark does not under the same parameterization. Parameter sensitivity analysis further illustrates how voting weights, decision thresholds, and the grand coalition worth interval are associated with changes in power indices and the distribution of allocation uncertainty. The numerical results provide methodological illustrations under a controlled parameterization and demonstrate the feasibility and computational implementation of the proposed solution procedure.