Assuming NP $\neq$ P, are there NPI languages only P languages reduce to?
I don't think such a language $L$ can exist, because it suffices to slightly modify it to get a language $L'\in L_c$ and not in $P$. 
For instance let $u$ be a word of minimal length which is not in $L$, and take $L'=L\cup\{u\}$. It is clear that $L'$ polynomially reduces to $L$: check if the input word is $u$, and if it's not use your oracle for $L$. Since $u$ is of constant size, this algorithm runs in polynomial time. So $L'\in L_c$.
But you also have the inverse reduction: $L$ polynomially reduces to $L'$. This implies that $L'\notin P$, because otherwise $L$ would also be in $P$.