No subset of RE captures the set of TMs halting on themselves as input
According to the recursion theorem, there is a Turing machine $M$ such that $L(M) = \{\langle M \rangle\}$. If $P$ is a property such that $L_P = K$ then $\langle M \rangle \in K$ implies that $L(M) = \{ \langle M \rangle \} \in P$. Now it is easy to construct a different machine $M' \neq M$ such that $L(M') = \{ \langle M \rangle \}$. Since $L(M') \in P$ and $L_P = K$, we must have $\langle M' \rangle \in K$, yet $M'$ doesn't halt on $\langle M' \rangle$.

Note that this solution doesn't use the condition $P \subseteq RE$.