Can alter-personas of a person with MPD/DID have mental diseases of their own that other 'alters' don't have Context: We are making a short film on different mental disorders.

The Question: If it is realistically feasible for a 'alter-persona' of a person with Multiple Personality Disorder to have different mental disorders of their own such as schizophrenia and depression.
Yes.  For example, people suffering from DID can also suffer from depression.

See DID

Since, by definition "dissociative identity disorder is characterized by the presence of two or more distinct or split identities or personality states that continually have power over the person's behavior. ... ", then the other disorder (e.g. depression) can be unique to a persona.