Battle Career experience bonuses are the new main reward for the brawl quests completed.
Battle Career experience bonuses do not spend the cap, and double the amount of Battle Career experience gained.
E.g. you have a cap of 1,000 points and 6,000 Battle Career experience bonuses.
You get 200 points for the battle. From the cap of 1000 POINTS we subtract 200 POINTS, now your cap is 800 POINTS.

Bonuses double the experience gained: you get 200 POINTS with bonuses. Subtract ONLY 200 POINTS from the number of Battle Career experience bonuses, and you have 6000 - 200 = 5800 bonuses left.

Thus, for a battle you will receive 200 POINTS + 200 POINTS = 400 POINTS for your Battle Career.  After the battle, your cap becomes 800 POINTS, and you have 5800 bonuses left.