Using closure properties to show that $a^n b^m a^{n+m}$ is not regular
That line of reasoning can work in principle; you are trying to use closure properties of REG for a proof of contradiction. Make sure you understand that that's what you're doing.

Unfortunately, you do not get $\{a^nb^n\}$. For instance, $a^3 b^4 \in \overline{L} \cap a^*b^*$. Check out our reference question for more techniques you can try.