It appears you are billing for the psychiatrist, and not the facility. You would use the same forms you always use, the 1500. The hospitalist and the psychiatrist are separate entities, correct? The hospitalist sees the patient every day; the other specialists see the patients as needed. You don't need a modifier.
The psychiatry section of the CPT book states that those codes are able to be used in all places of service. Therefore, if it appropriate, you are generally going to use the same codes you use when he performs those same services in an office setting (90785, 90791-90792, 90832-90840, etc). Make sure the POS code, the facility, the date of hospitalization, and the referring doctor are all included on the claim.
Since psychiatrists are MDs, they can also use the regular hospital codes if it is more appropriate (99231-99233), and if they are not doing psychotherapy, then I would think the regular hospital codes are indeed more appropriate.
In the Clinical Examples section of the 2020 CPT book, under 99232 (Subsequent Hospital Care... usually the patient is responding inadequately to therapy or has developed a minor complication...), we find this example: Subsequent hospital visit for female who continues to struggle with suicidal thoughts and has multiple questions about her antidepressant medication (Psychiatry Specialty).
One example for 99233 shows "... who is actively suicidal and experiencing significant medication side effects." (Psychiatry Specialty)