Get a gold medal in the indicated campaign mission to unlock the Great Battle. 

Basement Baloney (Great Battle 3)   : Campaign mission 2 Bathroom
Blitz (Great Battle 4)              : Campaign mission 3
Blind Pass (Great Battle 2)         : Campaign mission 1
Flooded Follies (Great Battle 5)    : Campaign mission 4
Sandbox Shenanigans (Great Battle 8): Campaign mission 7
Tricycle Treachery (Great Battle 6) : Campaign mission 5
Waterway War (Great Battle 7)       : Campaign mission 6