= Solution
Use the source action $S_{\rm src}=\int d^4x\,[J\phi+\bar\eta_\alpha\psi_\alpha+\bar\psi_\alpha\eta_\alpha]$, with independent <Grassmann variables> used as sources $\eta,\bar\eta$ and left <functional derivatives>. A scalar insertion is $(1/i)\delta/\delta J$. For the odd fields the ordered rules are
$$
\boxed{\phi\longleftrightarrow\frac1i\frac\delta{\delta J},\qquad
\bar\psi_\alpha\longleftrightarrow i\frac\delta{\delta\eta_\alpha},\qquad
\psi_\alpha\longleftrightarrow\frac1i\frac\delta{\delta\bar\eta_\alpha}.}
$$
The middle sign is not arbitrary: differentiating the rightmost odd source in $\bar\psi\eta$ from the left gives $\delta_L(\bar\psi\eta)/\delta\eta=-\bar\psi$. Derivative ordering must consequently match the displayed order of the fields; the rightmost derivative acts first.
Expand $\exp[ig\int\phi\bar\psi\psi]$ in its <power series>. Replace each field in every term by its source insertion, preserving the <fermion> order. These derivatives can be pulled outside the free <path integral>. Re-exponentiation gives the unnormalized <generating functional>
$$
\boxed{Z[J,\bar\eta,\eta]=
\exp\left[ig\int d^4x
\left(\frac1i\frac\delta{\delta J(x)}\right)
\left(i\frac\delta{\delta\eta_\alpha(x)}\right)
\left(\frac1i\frac\delta{\delta\bar\eta_\alpha(x)}\right)\right]Z_0[J,\bar\eta,\eta].}
$$
Summation over the <spinor> index is understood. Normalized <correlation functions> require division by the same interacting $Z[0,0,0]$, which cancels <vacuum> factors; that division is separate from this operator identity.
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