
(Picture is small, here is another link: http://i.imgur.com/OJC0A.png)
http://www.crummy.com/software/BeautifulSoup/bs4/doc/#strings-and-stripped-strings suggests that the .strings method is what you want - it returns a iterator of each string within the object. So if you turn that iterator into a list and take the last item, you should get what you want. For example:
$ python
>>> import bs4
>>> text = 'unwantedwanted'
>>> soup = bs4.BeautifulSoup(text)
>>> soup.find_all("div", style="mine")[0].text
u'unwantedwanted'
>>> list(soup.find_all("div", style="mine")[0].strings)[-1]
u'wanted'