Classification of the complement of a language
The complement of $L$ is:

$\bar{L} = \{\langle M \rangle \mid \text{ all strings accepted by $M$ have the same length} \}$

Your intuition that $\bar{L}$ is not recursively enumerable is correct, it is in fact complete for $\Pi^0_1$ by reduction from the emptiness problem. If you only need to prove that $\bar{L}$ is not in $\textsf{RE}$, it is sufficient to observe that $L$ is undecidable (reduction from the halting problem) but $\textsf{RE}$ (trivial).