Is informing the manager about a colleague at work who has a bad reputation and inappropriate behavior, and requesting a choice between her staying or my staying, considered a severance of her livelihood?

You must continue to advise your friend and keep her away from sins, by concealing her faults and not disclosing her secret.

As for your conversation with the manager, if it is for advice and the benefit of work, then there is no harm in it, provided that you limit it to what is necessary to reveal. Otherwise, it is not permissible if the motive is defamation or mentioning defects unrelated to work.

Regarding taking money from her salary, you are allowed to take only your due right without increase, and to ensure the consequences of your action. However, taking a right for others is not permissible.

As for giving the employers a choice between your staying or her staying, there is no harm in it if you are harmed by her presence.

After fulfilling your duty of advising, you are not permitted to delve into her honor, for Allah Almighty said: ﴿So turn away from them, for you are not to be blamed.﴾