Online Multiple Testing (OMT), a fundamental pillar of sequential statistical inference, traditionally evaluates the False Discovery Rate (FDR) and statistical power in isolation, obscuring the highly asymmetric costs of false positives and false negatives in modern automated pipelines. To unify this evaluation, we introduce $\textit{Weighted Regret}$. Under this metric, we prove the $\textit{Duality of Regret Conservation}$: purely deterministic procedures ensuring strict FDR control inevitably incur an $Ω(T)$ linear regret penalty, as threshold depletion during signal-sparse cold starts forces massive false negatives. Tailored for exogenous testing streams, we propose Decoupled-OMT (DOMT) as a baseline-agnostic meta-wrapper. By incorporating a history-decoupled, strictly non-negative random perturbation, DOMT rescues purely deterministic baselines from severe threshold depletion. Crucially, it preserves exact asymptotic safety in stationary environments and rigorously bounds finite-sample error inflation during cold-starts. Guaranteeing zero additional false negatives, it yields an order-optimal $Ω(\sqrt{T})$ regret reduction in bursty environments, with a derived ``Cold-Start Tax'' characterizing the exact phase transition of algorithmic superiority. Experiments validate that DOMT consistently curtails empirical weighted regret, achieving an order-optimal sublinear mitigation of threshold depletion to navigate the non-stationary Pareto frontier.