Edited by kmortis:Removed previously moderated content and response to same
If a person is suffering chest pains, you don't take a brief side trip to the homeopaths, you go to the emergency room.
A dangerously delusional person requires proper and immediate attention. Been there, seen that.
ETA: If after evaluation, a medical professional determined an exorcism might be a harmless placebo, meh who cares. But I would doubt such a "treatment" would be recommended.
Sorry for (removed by mod) insult, I misread your post.
A non-mental illness example might make my point clearer. If someone is suffering from a fatal but treatable illness (let's say it's a form of cancer with 80% surgery/chemo success rate) and someone you care about is refusing treatment, you talk to them on Saturday and they promise to go to the hospital and start treatment Monday if you will first take them to a priest on Sunday for the rite of anointing of the sick, and to the homeopathic store to pick up some so-called remedy, just in case.
If you refuse, they won't get medical treatment and will die very painfully. If you agree, they will have an 80% chance of survival and you will have made them happier regardless of outcome.
So if someone is suffering from mental illness that is at that moment acute, they are an imminent threat to themselves or others, that is one thing - analogous to your heart attack example. But if it's more chronic, they've had very poor compliance with treatment in the past, they've been released because not (normally) deemed a threat, and if your cooperating with them in seeking an exorcism first, if you care about them, why not, because it increases the chance of a successful long-term outcome by increasing chance of compliance.