If a boy becomes Bar Mitzva during Sefira, may he take a haircut for the Simcha - similar to a Bris where the father is allowed to take a haircut in Sefira?
No. The simcha of the barmitzvah is our joy in the boy becoming obligated to keep mitzvos and the Jewish way of life.
When will he have his haircut?
If he takes it after his birthday, we would introduce the barmitzvah boy to mitzvos by making him go against our practice which is surely not the right thing to do.
if he takes it before his birthday, what sort of chinuch - education is that for his adult life?
Nit'ei Gavriel (Pesach, vol. 3, 50:11) cites authorities on both sides of the issue:
Practically, as usual, CYLOR.