Hi Nonny, she did not poison him! Duke Bastien dies either way. If you give Vivienne the snowy wyvern heart, he will live long enough to say words of farewell to her. If you give her the wrong heart, the potion will poison him and Vivienne will not be happy, naturally.
I found it strange that giving him the correct potion would still result in his death, with little explanation from Vivienne either way, but later Bastien’s daughter says that he was suffering from a terminal illness and it “was simply his time.” If you’re her friend, Vivienne will give you a custom ring (not equipment, but a gift) for helping her.