Let $F_n$ be the Fibonacci sequence and $\chi$ a non-principal primitive Dirichlet character. Does there exist $n$ such that $\chi(F_n) \neq 0,1$?

One way to prove this would be to obtain non-trivial bounds for sums of the shape $\sum_{n \leq x} \chi(F_n)$.

It is foreseeable that there could be some "bad" Dirichlet characters where one does not obtain the result, so I'm very happy to ignore finitely many Dirichlet characters of any given order (say).

More generally, I'd like to know a version of this where $F_n$ is replaced by an arbitrary Lucas sequence.