1 - Go into MMB of your BAH reservation.
Ensure you select "go to previous design" at the top of the screen on either the website or the app version. The "new" version of MMB which is the default is particularly useless and not worth the time of day.
2 - Pick any given flight within the reservation and scroll and select the link for more flight information.
3 - You will see the booking class for the flight in question.
Repeat process for all other flights in the booking.