Key to thinking about the answer to this question is the motivation of the king's daughter. We are given key clues that help us identify more about the kind of person she is and based on that we are able to make inferences based on how she would decide the fate of her lover. Another thing to consider is how the author of this excellent tale gives us enough evidence to support both choices without indicating that one would be more unlikely than the other. Consider how he does this:
How often, in her waking hours and in her dreams, had she started in wild horror, and covered her face with her hands as she thought of her lover opening the door on the other side of which waited the cruel fangs of the tiger!
But how much oftener had she seen him at the other door" How in her grievous reveries had she gnashed her teeth, and torn her hair, when she saw his start of rapturous delight as he opened the door of the lady!
Both options are shown to be equally terrible to the princess, and so when we consider which she chose for her lover, we are left confused and unsure - evidence is provided to suggest that both are possible. How you decide to answer the question will probably say more about you than it will the story, for it is a masterful example of ambiguity.
I think she would choose the tiger because he said that she hated the lady behind the door.
Also she would hate seeing her lover with another lady.