Okay, I feel dumb... It's so easy I should have thought of it myself. It works, but now there is another problem: The interaction functions for the character have to be in the global script so I can't have the animation stop and make the cup object visible again before starting any conversations. If I don't, the character will either still be locked in the animation view, or if I don't lock it, she will change to the speech view and the cup will disappear because it is not set to visible yet.
@Onker:
The animation is eNoBlock. The problem was appearing when the player did something blocking, which includes speaking to the character.