The biggest red flag that I see is that this therapist "doesn't believe in drugs or BPD". Like he believes that BPD doesn't exist? And nobody should take any kind of psychiatric medication? I'm not saying that your wife does or does not have BPD or that she should or should not take medication. But most mental health professionals agree that there is a set of personality/mood traits that point toward BPD. And many people have been helped by psychiatric medications. To completely eliminate either of those as possibilities seems like malpractice to me.

It really does seems like marriage therapy would only work if both people felt like the therapist supported them. I also don't think that you have to accept her yelling at you. I can also understand that it would get very draining to listen to someone "vent" about hurt feelings/depression, especially if that venting never led to any change. All the more reason it seems like she needs individual therapy...someone else to support/listen to her and someone who could give her some ideas about how to improve things within herself.

Have you tried talking to the therapist about your view that he always sides with her? I wonder what his reaction would be.