As Pinja noted, a paper by Andersson et al. (arXiv)(DOI) has been especially useful. The paper goes into a great deal of detail, and I finally sat down today to take a proper look at it. As an example problem, I picked two qubits with an exchange interaction to check this which is a minimal version of what I'm considering. To begin, the master equation is given by

$$
\dot\rho = \Lambda(\rho).
$$

The method requires that basis operators of the system are chosen. It is convenient to give these in terms of the Pauli matrices in the case of two qubits, but for a qutrit one would employ the Gell-Mann matrices. Defining $\sigma_i = \mathbf{1},\sigma_x,\sigma_y,\sigma_z$ for each qubit, this system has a basis built up of the tensor products of these with a factor of $1/2$ for normalisation, yielding 16 operators $G_i$ e.g. $G_5 = G_{xx} = (\sigma_x\otimes\sigma_x)/2$. Sticking with Hermitian operators keeps things neat as well, since some daggers can be neglected.

A special matrix is now composed called $L$, which is related to the master equation.

$$
L_{n,m} = \mathrm{Tr}[G_n\Lambda(G_m)].
$$

If we are dealing with the master equation as a matrix acting on a vectorised density operator as discussed in the question, then this can be expressed as

$$
L_{n,m} = \mathrm{vec}(G_n)^\dagger\,\Lambda\,\mathrm{vec}(G_m),
$$

which allows L to be derived in a single matrix equation, but that's getting a little off topic.

In the sample case I considered, $L$ is does not contain time varying terms, so it may be exponentiated to get a new matrix $F$, which is related to the solution of the master equation $\phi$

$$
F(t) = \exp(Lt).
$$

$F$ can be used to get a Choi matrix $S$, which is exactly what I need. At this point, a basis needs to be chosen for the future Krauss operators. I'm quite happy with the Pauli operators so I'll stick with those for this next equation,

$$
S_{a,b} = \sum_{n,m}F_{m,n}\mathrm{Tr}[G_nG_aG_sG_b].
$$

Finally, the wonderful part.

$$
\rho_t=\phi_{n,m}(\rho_0,t) = S_{n,m}(t)G_n\rho_0 G_m^\dagger
$$

As you can see, $S$ is a matrix of weights for a sum of superoperators *in a useful basis* that I can select. This has been referred to as the *process matrix* (arXiv)(DOI) which is unique to a process in a given basis. In the sample case, in which the master equation has no time dependent terms on the RHS, the solution can be directly verified by representing $\Lambda$ in matrix form and exponentiating it to get $\phi(t)=\exp(\Lambda t)$.

This works in the time independent case for quits and qutrits as expected. I need to check that this works in the case of time dependence.

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