= Solution
Let $K=H-\mu N$, take $M$ slices of width $\epsilon=\beta/M$, and insert the <coherent-state resolution of identity> between factors of $e^{-\epsilon K}$. If $H$ is in <normal ordering>, each short-time matrix element is
$$
\langle\psi_{j+1}|e^{-\epsilon K}|\psi_j\rangle=\exp\left[\sum_n\bar\psi_{n,j+1}\psi_{n,j}-\epsilon K(\bar\psi_{j+1},\psi_j)+O(\epsilon^2)\right].
$$
Combining the overlap with the Gaussian weight leaves $\sum_{j,n}\bar\psi_{n,j}(\psi_{n,j}-\psi_{n,j-1})$ in the action. The <coherent-state time slicing> prescription fixes which adjacent labels appear in $H$, rather than allowing an arbitrary ordering change after taking the continuum limit.
The twist $\zeta$ in the thermal trace closes the path with the <coherent-state thermal boundary conditions>: periodic for bosons, antiperiodic for fermions. Taking the regulated continuum limit gives
$$
\boxed{Z=\int_{\psi(\beta)=\zeta\psi(0),\ \bar\psi(\beta)=\zeta\bar\psi(0)}\mathcal D(\bar\psi,\psi)\,e^{-\mathcal S},\qquad \mathcal S=\int_0^\beta d\tau\left[\sum_n\bar\psi_n(\partial_\tau-\mu)\psi_n+H(\bar\psi,\psi)\right]}.
$$
The action is dimensionless because $\tau$ has inverse-energy units. For fermions, $\bar\psi$ and $\psi$ remain independent <Grassmann fields>. If the original <Hamiltonian> is not normally ordered, first express it in normal order, retaining all constants.
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