Let's discretize time $t$. In other words, assume that we have $n$ Grassmann-odd variables $\theta^1, \ldots, \theta^n$. In this language OP's differential operator $\partial_t$ is replaced with some Grassmann-even matrix $A^i{}_j$. Define $$\theta^{\prime i}= \sum_{j=1}^nA^i{}_j \theta^j.$$
Then a straightforward calculation shows that OP's path-integral is replaced with $$\begin{align} \int d\theta_n \ldots d\theta_1~\theta^{\prime 1}\ldots \theta^{\prime n}~=~&\ldots~=~ \int d\theta_n \ldots d\theta_1~\delta(\theta^{\prime 1})\ldots \delta(\theta^{\prime n})\cr ~=~&\ldots~=~\det(A),\end{align} $$ which indeed is a discrete version of OP's claim (2).