A possible counterexample?:
           Triangle http://cs.smith.edu/~orourke/MathOverflow/BallEllipseTriangle.jpg
The tetrahedron is nearly a flat rectangle (red), and the ellipse $E$ nearly fills it. Then I don't see how to enclose $E$ in a triangle that remains in the ball. Not a proof, I know...

A possible counterexample?:
          
The tetrahedron is nearly a flat rectangle (red), and the ellipse $E$ nearly fills it. Then I don't see how to enclose $E$ in a triangle that remains in the ball. Not a proof, I know...