Diagnosing a personality disorder in someone who has a bipolar disorder is complicated. Personality disorder means an enduring and problematic pattern of behavior, especially in relationships with others. Theoretically, this disorder is distinct from mood symptoms like depression, mania, or hypomania. But a sharp distinction is only possible if the problematic pattern of behavior occurs when mood symptoms are completely resolved. Unfortunately, most people living with mood disorders are still waiting for the day when that happens. In reality, the theoretical distinction between personality disorder and a mood disorder is hard to find.
Diagnoses of personality disorder also have a complicated history. They were often seen as more psychological while mood disorders (like bipolar disorder) were seen as more biological. This view sometimes led to more blame and shame being attached to personality disorders.
Thinking about your relative, I will assume that the earlier diagnosis of bipolar disorder is correct. If someone has had a true manic episode, then we would make a diagnosis of type 1 bipolar disorder. True manic episodes are not symptoms of personality disorder, so the question is whether he has a personality disorder in addition to bipolar disorder.
Ultimately, diagnoses are only useful if they guide us to practical things that help. Our question regarding your relative would be what different advice we might give if the work and family problems are due to bipolar disorder or to personality disorder.
In either case, we would take a second look for other possible signs of mania or hypomania. Problems like unrealistic self-image, irritability, and impulsivity can certainly be symptoms of a personality disorder. But these symptoms more likely indicate mania or hypomania if they are accompanied by other typical mania symptoms, such as racing thoughts, rapid or pressured speech, and decreased need for sleep. If we see these other signs of mania or hypomania, then we would certainly ask whether medication treatment needs to be adjusted.
Whether we think work and family problems are due to a personality disorder or bipolar disorder, we would recommend counseling or therapy. Whatever the diagnosis or cause, working with a therapist to recognize and tone down problematic behaviors is likely to help. For psychotherapy, specific diagnoses are less relevant or useful. Therapy should focus more on what you do (being easily irritated) than on what you are (having a personality disorder or a bipolar disorder).