Generalizing Brzozowski's DFA minimization algorithm to finite automata with different classes of accepting states?
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).