Find a language $L$, such that $L \notin \mathbf{P}$, but $L^* \in \mathbf{P}$
Am I on the right track? It seems to me that I'm trying to prove something stronger than what I'm asked


That looks like an excellent approach. Proving something stronger is sometimes easier1 and usually useful.

What's particularly nice about your template for $L$ is that it opens up new questions about what happens if you generalise to $L = A \cup B$ where $A$ and $B$ are in different complexity classes and $B \subset A^*$. The extreme of $A$ regular and $B$ undecidable is good enough, but you could also consider $B \in NP^c$, $B \in EXP^c$, maybe specific context-free $A$, ... Perhaps it would be an interesting question to ask how close they can be to each other while still getting some kind of separation. If you enjoy this kind of thing, there may be a mini research project in it.



1 This is particularly true of proofs by induction, where proving a tight bound often makes the inductive step work where a loose bound wouldn't.