I think it also depends on what kind of psychologist you want to be. Do you want to do clinical practice or research? I know PhD psychologists who work in academia as researchers in related fields to actually create new forms of patient education, therapy, etc. But they don't actually see patients in a clinical setting. I currently work with a social psychologist who doesn't see patients but conducts research on healthcare education initiatives using behavioral learning methodologies. Psychologists are not just clinicians. So I think you can definitely have a mental health condition and be a psychologist. It may take you a bit longer to finish your PhD, and you may need to adjust your practice to handle the work load and self care.

Good luck!