In the typical constructions of the path integral I have seen, they always start from the assumption that the Hamiltonian one starts with is a function of $x$ and $p$, aka for $n$ degrees of freedom:
$$H=\sum_k^n \frac{p_k^2}{2m_k} + V(x_1,x_2,\cdots,x_n) \, ,$$
which allows one to utilize Trotter splitting to simplify the propagator. In the derivation of the path integral, this step is at
$$ \langle x_f |e^{-iHt/\hbar}|x_0\rangle = \int dx_1\cdots \int dx_{N-1}\prod_{k=1}^{N}\langle x_{k}|e^{-iH\Delta t_k/\hbar}|x_{k-1}\rangle$$
where you can write in the limit that $N \rightarrow \infty$ $$e^{-iH\Delta t_k/\hbar} \approx e^{-iV\Delta t_k/2\hbar}e^{-iT\Delta t_k/\hbar}e^{-iV\Delta t_k/2\hbar}$$
where the potential terms are now directly attached to position states, leading to nice things down the derivation.
How does the derivation change, and thus the path integral formulation change once you no longer assume you start with a Hamiltonian that depends on position and momentum?
Let's say there is a two state Hamiltonian $H = A\sigma_x + B\sigma_z$ (where $\sigma$ are the Pauli spin matrices), and now the Trotter splitting no longer makes sense. How does one proceed with calculating the propagator from here?