Can't find a mistake in reduction from RE language to a non-RE language
Your reduction actually shows that $L_m$ is not co-RE. Indeed, if $L_m$ were co-RE, then your reduction shows that $L_u$ is also co-RE. However, $L_u$ is known to be RE but not recursive, hence cannot be co-RE.

Similarly, you can show that $L_m$ is not RE, by reduction from the complement of the halting problem: given an instance $(M,w)$ of $L_u$, create a new machine $M'$ which on input $n$ simulates $M$ on $w$ for $n$ steps. If $M$ halts, $M'$ gets into an infinite loop, and otherwise it halts. You can check that $M$ halts on $w$ iff $L(M')$ is finite.

The language $L_m$, usually known as INF, is actually one of the canonical examples of $\Pi_2$-complete languages, making it harder than the halting problem.



Your title suggests that you believe that RE languages cannot be reduced to non-RE languages. But in fact, every recursive language can be reduced to every non-RE language. I'll let you work this out.