A billiard trajectory inside an ellipsoid in $n$-dimensional space is tangent to $n-1$ quadrics confocal with this ellipsoid.
In your case, the disk bounded by $C$ is a limit of ellipsoids confocal to $E$, lines intersecting $C$ play the role of tangents, so a line starting at $C$ returns to $C$ after one reflection.
The proof of the general result above can be found in
Tabachnikov, Serge, Geometry and billiards, Student Mathematical Library 30. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3919-5/pbk). xi, 176 p. (2005). ZBL1119.37001.
The idea is to use ellipsoidal coordinates as Jacobi did when studying geodesics on ellipsoids. Billiard trajectory inside an ellipsoid is the limit case of a geodesic on an ellipsoid one dimension higher (as the higher-dimensional ellipsoid flattens, a geodesic going over the "edge" becomes billiard trajectory).
The argument in the book cited is for ellipsoids with different half-axes. The general case is proved by going to the limit. As two of $\lambda_i$ approach, some of the quadrics from confocal family degenerate (to double planes, I guess, so that a trajectory whose first segment lies in such a plane, always remains in the plane). But here we are interested only in those quadrics from the family which are ellipsoids. Let $\lambda_i(t)$ be all distinct for $t \ne 0$ and some of them coincide for $t=0$. For every $t$ consider the corresponding ellipsoid $E(t)$ and its confocal family. The billiard trajectories inside $E(t)$ depend continuously on $t$, and so do the ellipsoids confocal to $E(t)$. Thus one can take limit as $t \to 0$.