Kiwi,
Unfortunately, we have very limited control over what the Fan Box area looks like. Facebook used to allow a custom CSS stylesheet where we could change everything however we wanted. They removed that feature about 2 years ago as it was heavily abused to remove all indications of "Facebook" and the branding that FB wanted to at least have when the widget is used.
So, there are really only 'big' options you can use here which may not be wanted, but it's all I can recommend:
1) Increase the width of the sidebar (I know you said that's not possible, but I have to say it)
2) Hide the stream altogether with the parameters in the JFBCFan widget.
That's really all that's possible. I wish there were more options, but I hope that helps at least explain the situation even if it doesn't really help fix the problem.
Thanks,
Alex