FPS collider center offsets camera pivot?

I just set up the standard character controller script and as by default the center is 0,0,0, but I need my "eyes" (camera) to be where the head is, meaning that I should move the collision capsule down.. So if my players height is 1.8 meter and his eyes are at 1.7, in the character controller inspector I put center offset Y to -0.8, but in the game, the camera (with mouselook script) pivots around the capsule center, and not around itself. So, not only it gives me ability to lean forward when looking down and see through walls. but also makes physics errors like jumping enormous heights when looking around...

I could get over this by keeping center and putting collider height to be 3.6 meters but then I can't go through doors =(

How do I fix this?