I doubt that diving has any direct biological effect on mental illness or the drugs used to control such illness; psychiatric drugs usually have long half-lives so they are not likely to wear off abruptly like antihistamines.
The question posed here, though, is one with broad social consequences; the answer relies so heavily on how severe the illness is, how controlled it is, what type it is, that no serious answer to it can be given in any broad terms.
One might as easily ask: would you let a schizophrenic babysit your kids? The answer could range from 'certainly' to 'are you kidding?' depending on the degree of control that has been established.
I am a little concerned that the patient here feels that interacting with the therapist is a painful experience akin to prostate exams. That is not a good sign. Such "private" patients may not like to admit that they aren't taking their meds or ar hearing the voices again, feeling such things are "private" matters. There can really be no privacy between a psychotic patient and their therapists. Unless the therapist knows exactly, and honestly, what is happening in that person's life, the degree of control cannot be established. And that control, frankly, separates a schizophrenic in a productive life from one in an institution. Cooperation fully with the therapist is not a personal option.
Remember, what we are talking about here is not a ruptured eardrum, but a psychotic disease. By it's definition, psychosis blurs the line between what is real and what is not. Some psychotic patients, like Alzheimer's patients, can only function very well so long as their environment is a very familiar one. Physicians know well the condition of "sun downing", when an elderly person with perhaps some borderline dementia is admitted to the hospital and then, in the evening (when all family goes home), suddenly decompensates and becomes flagrantly disoriented and combative. Suddenly removed from the home environment, an undetectable degree of dementia becomes fulminant.
I worry that the same thing may happen at depth, in the sensory isolation of the ocean. In this unfamilair world, alone with our thoughts and the sound of nothing but bubbles, might a well-compensated psychosis suddenly decompensate? That diving buddy means to kill them, that barracuda must be attacked, this regulator in my mouth is choking me, put there by the devil. I don't mean this in any humorous way at all. That's how a schizophrenic person can think when their control fails.