Why be pessimistic? I think it would be wonderful for you and your friend to be together for Pesach! True, there is a technical matter of your not having dog food that is chametz in your domain during Pesach, but it's not really a major problem since it is not your chametz and you are not taking any responsibility for it - and you cannot destroy it on Pesach against your friend's will.
Why not simply explain how much you'd like her to come so you could spend Passover and the Seder together, but that you have to be honest and tell her that you feel uncomfortable having chametz in your home during Passover? However, why not tell her that you would be more than happy to do whatever you can to provide gourmet "Kosher for Passover" meals for her dog? You are hosting her and you are her friend and would very much like to be together - even if it costs you something to buy kosher for Passover dog food. And who knows? Maybe her dog is "Jewish" and will love the chicken soup and matza balls.
I hope you both (actually all three of you) are able to share a truly joyous, kosher and inspiring Passover.