 Solution

Under $\Psi \mapsto \Psi +\delta \Psi$, the transition amplitude changes to first order by an insertionIfwith the corresponding condition on the bra, the two terms vanish after moving $\hat{Q}$ to the external states. The amplitude is then independent of the gauge-fixing choice.

Conversely, invariance for arbitrary changes of the gauge-fixing fermion requires physical external states to be BRST closed. States differing by a BRST-exact state have identical matrix elements against closed states, so the physical state space is the cohomology