This is correct. To expand on this, you say they are in the inpatient unit 4x a month laughing, reporting psychotic symptoms in a rote fashion using psychiatric jargon, leaving when their check comes out (and presumably denying that they are a safety risk), and showing no objective signs of mental illness? Make sure your inpatient team DOCUMENTS THIS!! Don't have a series of daily progress notes like "Reports ongoing AH, VH, +SI, mood despondent, affect flat, r/o bipolar, r/o schizophrenia. Remains high risk appropriate for inpatient." All too often I see inpatient providers doing this, writing notes that make the person look legitimately ill. When the past ten admissions' notes make it look like the person is seriously mentally ill, it becomes harder to deal appropriately with the situation. So start by documenting accurately, and making sure others do the same.
Next, enact an appropriate plan when they show up again. Personally I think saying "I don't believe you. If you really wanted to kill yourself, you would have done it by now" is rather callous. It sounds almost taunting, and I would avoid emotionally charged "call-outs" even if it may feel satisfying. Presuming that there is at least some doubt about the malingering (which typically there is), I often use something more along the lines of: "Mr. X, you have been admitted eight times over the past two months, and looking over the records it doesn't seem like you benefit from these admissions. I also see that you haven't been to see the outpatient psychiatrist we referred you to. At this point, I think continuing to admit you to the hospital may actually be making things worse." Follow up with a pep talk about how being connected with outpatient management offers the best long-term prospect for improvement, and counsel them about drug use and other factors that may be worsening their mental state. Give them contact information for substance abuse programs if indicated. You then need to hold firm to the discharge plan, informing them that you are discharging them from the ER with a plan for outpatient followup. Document this very carefully, making sure that you explain the risks and benefits of your chosen management plan and showing that you exercised medical judgment in doing so. If there is a bad outcome, you want it on the record that you exercised medical judgment and chose the most appropriate treatment plan for the patient.
If they are straight malingering, and you are very confident, you can also document that and say that psychiatric followup is not indicated. I would avoid taking this lightly, though, and make a compelling case in your note.
Having had these interactions with many patients, in my experience when they see that they are not getting what they want they often want to just wrap things up so that they can go to another emergency room and get admitted there. Once in a while, things get uglier and they need to be escorted out by security. If that is the case, so be it. Odds are they will end up admitted somewhere if they really want to be, but you can still do what your medical judgment tells you is the right thing.