As I’ve mentioned before, Kurt Vonnegut once said that every character should want something, even if it’s just a glass of water. Desires drive character.
But there’s another side to purpose beyond desire. The other side of purpose is function. What’s the character good at? What’s the character made to do?
So check back to the character’s intersecting opposites. What do those opposites tell you about what the character wants? Or what do those opposites uniquely qualify the character to do?
The big-hearted, tough dude might just want to keep his friends safe, and be good at it too.