Implementing Descartes' Theorem

No you cannot apply Descartes' theorem for the circles with radius $9$, with radius $3$ and line $CD$ to find the radius of the cyan colored circle, as $CD$ is not tangent to the circle with radius $9$.

On your question of alternate approach to find radius $r$ of the cyan colored circle -

If $O, P , Q$ are respectively the centers of circles with radius $9, 3$ and $r$, note that

$PQ = 3 + r, PE = 3 - r$

Also, $OQ = 9 - r , OE = OC + CE = 3 + r$

Now applying Pythagoras theorem, $QE^2 = PQ^2 - PE^2 = OQ^2 - OE^2$

That gives you value of $r$. Once you know $r$, you can find the distance between identical circles which you mention you already know how to solve for.