Determining the classification of languages
Here are two hints


If $\bar L$ would be decidable (recursive), then so would be $L$, and you just argued that $L$ is not even recursively enumerable. Maybe you have a closer look on what $\bar L_0 \cap \bar L_1$ really is. For example
$\bar L_0 = \{ \langle M, w, 0 \rangle \mid \text{$M$ does not halts on $w$}\} \cup \{ \langle M, w, x \rangle \mid x\not=0\} $.
If you want to argue formally correct that a language is not recursively enumerable, or decidable, then try to use a reduction, for example the many one reduction. Reduce  a a problem, whose status you know, to $L$ and you can say something about $L$.