General question about IV
In general, if $c_1$ is endogenous, you need instruments for $c_1$ as well.

Example: Even when $x_1$ is exogenous, if $c_1$ is endogenous and $x_1$ and $c_1$ are correlated, then OLS (which is the IV estimator using $x_1$ as IV for $x_1$) is generally inconsistent.

Exceptions exist. For example, when exactly identified, if $(Ezc') (Ecc')^{-1} (Ecu)=0$, then your IV estimator is consistent, I guess. (Can derive this using standard technique involving the law of large numbers.)