So, I didn't end up finding the condition in the references that you suggested, but they were very helpful in helping me locate a book from with the condition I wanted could be easily derived, so I'll accept your answer (especially since these are very good references in general). The ``two cycles'' condition can be easily derived from Theorem 1 in Section 16 of Markov Chains with Stationary Transition Probabilities by Chung.

Thank you very much for the references! As you say, the first reference has more than I could ever hope for, so It'll take me a while to parse. Do you happen to know if the ``two cycles'' condition that I mention appears anywhere?