No, you can’t continue to manage this painful situation. Neither can your girlfriend. Her illness has become the most important “person” in your relationship. I’m sure this was not her intention. But she is more involved with her illness than she is with you. Quite inadvertently, and with the best of intentions, you have supported her in being ill by being as scared as she is. Talking with her in detail about her desire to kill herself isn’t helping either one of you.
A session of therapy every other week is clearly not enough for this level of distress. I suspect that her therapist doesn’t understand the intensity of your girlfriend’s symptoms or she would have increased the frequency of sessions long ago.
My best suggestion is that you join her in her therapy sessions for awhile — or at least for part of her therapy sessions. If you attend even some part of the sessions, you and the therapist can focus on getting you out of being a crisis worker and back to being a loving support. Ideally, you will change the situation so you are partnering with the side of your girlfriend that wants to be well instead of the side of her that is afraid of change. The therapist can help your girlfriend shift her need for crisis intervention to someone who is trained to manage it. Her therapist, not you, should be assessing her need for an inpatient stay.
Your girlfriend doesn’t want to lose you. Using you for daily crisis sessions isn’t the way to “keep” you. I think and hope that if you are clear and loving in insisting that you participate in helping her heal in a useful way by working together in therapy, she will go along. If not, by all means call your local Emergency Services Team to assist you when she talks about suicide.
I wish you well.