i m a psych resident specializing in neuropsychiatry so this is my perspective. behavioral neurology (its behavioral neurology if you're a neurologist and neuropsychiatry if you're a psychiatrist) fellowships pretty much exclusively focus on the dementias. the few in psychiatry departments have a little broader focus but this is to include the psychogenic or conversion disorders like fugue states, psychogenic movement disorders etc. but you wont get much if any expsorue to the primary psychotic or mood disorders. also they are 1 year fellowships than can be extended to 2 years for a year of research typically. some are 2 years but only 1 year is clinical. you can use your elective time to do more psychiatry rotations during residency.
there is actually a very expansive theoretical basis to psychiatry though most of it isn't taught during residency unfortunately. psychiatry draws upon social sciences, psychodynamic theory, cognitive, developmental and systems neuroscience, as well as evolutionary and molecular and cell biology. the basic functional neuroanatomy of emotion and cognition are fairly decently described (though this is an emerging field) and the neuroscience of consciousness is also something that is in development. though the theoretical basis of neurological problems is better established in general than psychiatric disorders, to my knowledge neurologists arent more effective at treating their disorders than are psychiatrists because despite all the advances in our understanding of neurobiology and genetics, neurologists still manage some fairly devasting degenerative diseases. as a psychiatrist i would argue there is a sound theoretical basis to how i conceptualize and manage patients and i routinely perform neurological examinations and look at neuroimaging on my patients. It is just i also have an appreciation of social, psychological, cultural, legal, and ethical dimensions of care in a way that is often more relevant than the biological.
the good news is that the stigma of psychiatric disorders means that the more affluent will see neurologists for their serious psychiatric disturbances including primary mania and psychosis/"schizophrenia" and most neurologists have no interest in working with these patients so you could certainly carve out a niche for yourself. just make sure you avail yourself of the opportunities to get further training and supervision in this and realize that the medical component is the least important in the management of patients with serious psychopathology.