Can this be translated (not just glossed, but actually translated) as "This is my brother Bashir, who is busy"?
Not exactly. The logical meaning is different between conjugation (connecting sentences) with "and" and "who" in English, and it's kind of similar in Arabic. (Who: الذي/m).
And was not a choice in the Bubbles
and he is = وَهُوَّ
how would be the feminine مَشْغولة؟