I see you are UK based. You might be interested to read about the work and approach of Rufus May, if you aren't already familiar with him. He is a clinical psychologist who was diagnosed with schizophrenia and he advocates non-medicinal approaches to managing the symptoms of psychosis. From memory, he doesn't advocate psychoanalysis specifically, but certainly believes that psychosis can be treated outside psychiatric services.

I think the ideas that psychosis is too complex to treat with anything other than medicines or that people experiencing psychosis are too unwell to be in alternative therapies are outdated. When I was in a period of recovery from psychosis, I started Jungian influenced therapy. My therapist would often say that psychotic clients can successfully engage in therapy since many delusions have their roots in archetypal thinking. Delusions have their own sense and psychotic clients are capable of accessing that meaning, indeed are probably best-placed to access that meaning when unwell.

Rufus May – Holistic approaches to mental health
Rufus May - Wikipedia