We investigate model risk and distributionally robust optimization (DRO) under marginal and martingale constraints. Building on our previous work, we address the previously open case of static hedging with second-period maturity vanilla options and hedging strategies involving a vanilla payoff. We also extend recent sensitivity results to settings where admissible models must satisfy a martingale coupling constraint. Our approach relies on a weak implicit function theorem argument to construct families of measures satisfying the prescribed constraints. We derive closed-form sensitivity formulas and characterize the corresponding hedging strategies when the underlying process is real-valued.