If I score both battlefields in my turn, pass, and then my opponent conquers the battlefields and pulls my unit into a battlefield with Charm which I win, do I score the point on that battlefield during my opponent's turn?

Ruling: Yes, you score the point on the battlefield during your opponent's turn when you win the battle.

Nuances:

• If it would be the final point needed to win, you cannot win because you haven't scored at all battlefields during "this" turn (your opponent's turn), so the game would draw instead.