The Ghanaian side lost 67-73 to Mark Mentors at the National Stadium in a round that had defending champions Kano Pillars beat Gombe Bulls 75-60.
Pillars remain unbeaten in the competition which ends tonight as Braves take on the powerful Kano side at 15:30 GMT.
Braves battled from a three point deficit (39-42) at half time to lead 50-49 at the end of the third quarter but the final quarter took another twist.
Customs fell in the fourth quarter to lose by six points and hamper its chances of progressing to the main continental championship.
With Pillars well on its way to another appearance at the main African Championship, Braves is in a straight fight with Gombe for the zone’s final spot.
Bulls play Aspac of Benin today at 13:30 GMT following Civil Defence tie against Mark Mentors in an all Nigeria fixture.
By Yaw Adjei-Mintah
@YawMintYM on Twitter