Would Iowa have changed the Battle of North Cape?

If an Iowa-class battleship replaced Duke of York at North Cape, would Scharnhorst have had a better chance to escape?

No. Scharnhorst's lost radar, the Allied radar-directed gunfire, and the destroyer screen still made escape very unlikely.

Scharnhorst was unlikely to escape regardless of whether Duke of York or an Iowa-class battleship headed the British force. Belfast had probably disabled Scharnhorst's radar earlier in the day, leaving her badly disadvantaged in darkness. Either Allied battleship could use radar-directed fire while destroyers closed to prevent a clean withdrawal. The counterfactual turns less on nominal battleship performance than on the information and containment already shaping the action.