A possible counterexample?:
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<img src="http://cs.smith.edu/~orourke/MathOverflow/BallEllipseTriangle.jpg" alt="Triangle" />
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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...