You cannot use duplicates (or Iron Mines, etc.) to save Renly if any opponent controls a King character.
The Eyrie will still work though, because "cannot be killed" is not a save effect.
You need to trigger Renly's ability right away when you draw a card using insight, before you add it to your hand. (Source)
- Renly's ability works with any insight, not just his own (e.g. Butterbumps).