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.
The animation is eNoBlock. The problem was appearing when the player did something blocking, which includes speaking to the character.