By your own admission, all you really know about bipolar disorder is the sensationalized perspectives of the media and TV. In essence, I feel, you are discriminating against this girl for reasons you do not even understand, and, perhaps worse than that, you are instilling that prejudice into your daughter. Have you ever seen this child harm anyone? How is it that you know this child is not medicated? How do you know that it is bipolar disorder potentially affecting this girl's moods?
There are many types of bipolar disorder -
-- At the extreme of the "up" scale, some people believe that they are invincible, to the extent of trying to fly off tall buildings, but at the other end of the scale others merely become increasingly witty and feel that they have more energy and a more positive view on all the little pleasures in life.
-- At the "down" side things become a little more complex. At the EXTREME some can have homicidal thoughts, others are simply grouchy to a level no worse than a woman with PMS! Some become sullen, depressed, and sensitive emotionally.
-- The speed at which the change from "up to down" occurs are also variable. Some bipolar sufferers 'cycle' every three years, others cycle over a few weeks.
I am a rapid cycle, depressive bipolar. When I am/was up, I felt like I was the funniest person on the planet - when I am/was down, I would sink into depression and be very tearful, to the extent that I attempted suicide many times before I was diagnosed, and I would have days of being extremely angry for no apparent reason. As a rapid cycler, I tend to go through a full cycle in about 6-8 weeks.
Medications for bipolar are tricky to establish, as different combinations of moods and different bodies require different approaches. It took about a year to find a good balance of medications for me, and it is still not absolutely perfect. This is an even more difficult task in children, because the medications suitable for adults are not approved for children under 18.
I would recommend that you talk to the girl's mother about your concerns, but before doing so do some reading up on childhood bipolar disorder - I am sure you can find plenty of information through a Google search - and make sure that the girl's actions are not simply childhood tantrums that you would not condone.
Also, when you speak to the girl's mother, I would tell her that you would appreciate them calling before allowing the girl to arrive on your doorstep, as it is not always suitable for her to simply turn up. It is a common courtesy, and one that you have a right to.