With feedback from my on-track coach, the lean bike, and some expert CSS coaching, I improved my body position each CSS level I took.
The lean bike really isolated the problem, that I was turning my body to the outside of the corner for some reason. The lean bike and coach encouraged me to open up my hips to the direction I was turning, and after a few circles it worked. It felt like such an extreme position because my body was habitually in a bad position, but it doesn't look extreme and certainly is more effective to ride!
The best part of getting my body position 'fixed' was I could now use my outside leg to really clasp onto the tank, instead of wearing out my inside leg trying to hold up my body weight and all the subsequent effects of that (wrenching the handlebars for support, and thereby adding steering input through the corner and making the bike unhappy).
I love the lean bike!