I'm following Peskin & Schroeder p.302, and am trying to show that the generating functional for the Dirac field can be written as
$$ Z\left[\bar{\eta}, \eta\right] = Z_0 \exp \left[-\int d^4 x d^4 y \bar{\eta}(x) S_F(x - y) \eta(y)\right] $$
starting from
$$ Z\left[\bar{\eta}, \eta\right] = \int \mathcal{D}\bar{\psi}\mathcal{D}\psi \exp\left[i \int d^4 x \left[\bar{\psi}\left(i\gamma^\mu \partial_\mu - m\right)\psi + \bar{\eta}\psi + \bar{\psi}\eta\right]\right] $$
I've performed the following shift
$$ \psi(x) \to \psi'(x) - i\int d^4 y \;S_F(x -y )\eta(y) \\ \bar{\psi}(x) \to \bar{\psi}'(x) + i\int d^4 y \; \bar{\eta}(y)S_F(x - y) $$
But my issue is with the following term that arises from this shift
$$ -\int d^4 y \bar{\eta}(y) S_F(x - y)\left(i\gamma^\mu \partial_\mu - m\right)\psi'(x) $$
I know that I can move $(i\gamma^\mu \partial_\mu - m)$ over to the left using integration by parts to get
$$ \int d^4 y \bar{\eta}(y) \left(i\gamma^\mu \partial_\mu + m\right) S_F(x - y) \psi'(x) $$
Since $S_F(x - y)$ isn't a Green's function of $\left(i\gamma^\mu \partial_\mu + m\right)$, but of $\left(i\gamma^\mu \partial_\mu - m\right)$, I don't see how this can reduce down to $-\bar{\eta}(x)\psi'(x)$. I've looked at this post, but I don't understand the argument given there. They state that $S_F(x - y)$ is invariant under $x \leftrightarrow y$, but surely that can't be right? I know that for the scalar field this type of shift works out nicely because you have $\left(\partial^2 + m^2\right)$. So you have to do integration by parts twice, which doesn't change the sign in front of $\partial^2$. But for the Dirac field here I don't see how to progress.
