Thanks, Sour Lemon, I really appreciate your feedback. One thing didn't describe about this situation was that the patient was alert, oriented, and did not even appear upset (it is hard to convey tone of voice and body language online). So when he said "I'm going to commit suicide if I don't find my dog" it did not strike me as credible at all. Instead, he came across as demanding. Contrast that with another patient a few years ago who tried to AWOL because he was worried about his dog that was alone in his apartment with no one to care for him. That patient was genuinely distraught, and in that case the social worker contacted his clinical team and made sure that his dog was taken care of. Because of the demeanor of the patient with the "lost dog" my judgment was that either 1) He was making it up, or 2) He might really have an issue, but since he didn't seem upset, I thought it would be better to help him find his own solution by contacting family/friends, like the saying "give a man a fish you feed him for one day, teach him to fish you feed him for a lifetime."
I am reminded of another incident that happened a few years ago when a patient told me that he needed to be discharged because his infant daughter was in the ICU. When I informed the psychiatrist, he told me to call the children's hospital and ask if they had any infants with the name the patient had given me. When I did so, the hospital said they had no infants with that name. When I told the patient this, he said "It's harder to get out of this hospital than I thought". So maybe the bigger issue is distinguishing between truth and fiction. How would you verify the "lost dog" story?