OPINION? Does the climax NEED to be a big fight?

Definitely not. Of course, most authors choose to have a big fight as their climax, but that's just because the main antagonist and conflict are often external forces, usually people. As long as in your climax, the protagonist comes face to face with the main conflict, you'll be fine. All stories have a climax, even ones for children. Do you see mive, gory fights in those? If not, then it's perfectly okay to have a climax that isn't a huge fight.