You missed a tiny detail when defining $C$: all those languages are RE by assumption; no other language can be an $L(M)$! In fact, $L$ is the index set of
$\qquad \{ L \in \mathrm{RE} \mid L \subseteq L\bigl( 0(0+1)^* \bigr) \}$
which is neither empty nor universal, so $L$ is not decidable by Rice's theorem.