Rotating character controller's collider

I've created a demo of a character running around a globe, but the fellow's character controller collider doesn't rotate with him, so my isGrounded check fails and he accelerates to mad high falling speeds while wandering across the equator. I've tried rotating the collider and controller via their local transforms without results, and my cursory searches indicate that's what I should expect.

Is this by design, or is there a way to actually rotate the CharacterController's collider about something besides the Y axis?

I've tested using plain colliders and colliders with rigid bodies and it works, but there's a lot of work I'd be doing tweaking physics to look smoother, or in the case of just using colliders, redoing CharacterController's collision/step/movement logic. I'd like to avoid that if at all possible :).