Generalizing Brzozowski's DFA minimization algorithm to finite automata with different classes of accepting states? Brzozowski's algorithm for converting a DFA into an equivalent minimum-state DFA is remarkably simple: if $R(D)$ denotes the NFA formed by reversing all the edges in a DFA $D$, making the old start state an accepting state, and making the old accepting states start states, and if $P(N)$ denotes the result of applying the subset construction to the NFA $N$, then $$P(R(P(R(D))))$$ is a minimum-state DFA with the same language as $D$.

We can think of a DFA as a computational device that accepts an input string $w$ and then outputs 0 if $w$ ends in a rejecting state and 1 if $w$ ends in an accepting state. A natural generalization of DFAs associated each state in the DFA with some natural number between 0 and $k-1$, inclusive.

To the best of my knowledge, it's possible to minimize these modified classes of DFAs by using a distinguishability-based minimization algorithm, such as the canonical one by Hopcroft. However, I can't see how it would be possible to adapt Brzozowski's minimization algorithm to this new class of automata because the key step (reversing the automaton) no longer has a clear interpretation in this generalized setting.

Is there a known generalization of Brzozowski's algorithm for minimizing these sorts of automata? If not, are there any theoretical reasons why we'd expect that such a modified algorithm wouldn't exist?
The answer to your question is yes.

See Bonchi, Bonsangue, Rutten and Silva's papers Brzozowski's algorithm (co)algebraically (shorter conference version) and Algebra-Coalgebra Duality in Brzozowski’s Minimization Algorithm (longer journal version with more generalizations). 

They give a (lightly) categorical presentation of Brzowzowski's algorithm, and use it to derive versions of it for more general classes of automata, including Moore automata (which gives an affirmative answer to your question).