You didn't say where that cousin lived and how observant that family was. That's important.
For example, in metro NYC at the time, the date could hinge upon the availability of the synagogue as there might be other boys with similar birthdays. These children were from the BabyBoom. A family might not wanted to share a date with other boys.
More observant families would more likely have the service the week of the birthday.
I never saw a bar mitzvah announced in a New York paper.
Things would be different in each area.