I’ll answer you point for point ok?
This first point - if you can postpone a birthday for exams, then surely you can postpone it because one person feels strongly about something?
As for the brothers, think of it like someone dying in the family (that is how shias view it - only it is about the ruthless execution of the highest family of all, that of our beloved Prophet (saw) ). If someone died in your family around the days of your birthday, would you celebrate your birthday in that month? Or would you postpone any and all celebrations?