This is related to Sum of Fibonacci sequence evaluated at a Dirichlet character, but can be also be considered as a stand-alone.

I did an exhaustive search on non-principal (not necessarily primitive) Dirichlet characters $\chi$ of modulus $\leq 12$ such that $\chi(F_n)\in\{0,1\}$ to get a better understanding of the "bad" characters in the quoted question. None was found. Maybe more computing power can help here.

Q: Can you provide a non-principal Dirichlet character $\chi$ of moderately small modulus (such that it still can be handled by a human for the above purpose) such that $\chi(F_n)\in\{0,1\}$ for all Fibonacci numbers $F_n$, $n\in\mathbb{N}$.

Of course, I would also be interested in a primitive example, in negative results for certain moduli or in cleverer approaches to construct/find such an example.