Can I become a mental health nurse if I am bipolar?

Yes. Provided you have been stable on medications and continue in ongoing treatment. Stress and boundary issues can emerge, but with good treatment most people with bipolar can live normal lives. Discuss with your physician and therapist. I would advise therapy throughout the process to deal with the unique stress of treating people with mental illness.
Bipolar nurse. Stable bipolar, like any other illness, cannot be discriminated against, as long as you are good at what you do.