How does one get rid of a trouble making union employee that is always lying about a supervisor?

What you need is documentation. Keep a journal. Write down what he/she said and the time and date of each occurrence. Gather what you need till you think you have enough to report to someone who can do something about it.

Because of legal issues, some businesses don't fire bad employees, but rather offer them a different position (one they know this person doesn't want) in an attempt to make him/her leave voluntarily.

Ask the employee to provide evidence to support what he/she is saying about his supervisor. If there is no evidence, give them a warning and if he/she does it again, give a final warning or show them the door.

But would it be a legal thing with the union if she keeps it up? Because its been going on for over a year.