The reason it is called the Bouncy-Ball is because the entire ship is basically a giant, flying, bouncy ball with lasers.
The Bouncy-Ball was assigned to play a leading role in Operation: Firestorm by attacking a planet that would be used as a strike base for future assaults. It shot at the planet, but was then shot down itself by surface-to-air missiles. Inv thought the ship was going to be destroyed, but Shockwave revealed the ship's secret weapon: it could bounce. The ship started bouncing around the surface of the planet, crushing humans in the process. When the ship finally stopped, DALs under Masaharu came out and cleared a sufficient area for the Shattered Mirror II to be teleported in.
The Bouncy-Ball was then used as a sort of HQ by the SMU forces in the normal universe until Blackout forced their surrender. After that, the B-B was forced to run away. D-Void then appeared and took over the ship by turning the bridge crew into host bodies for himself.