Update and likely solution!
Figured I'd give this sunken thread a little update.
I've been experimenting with a few things and I think I found an acceptable solution!
There's a supermarket in my area called Tony's Fresh Market, and they sell select boxes of Finish Advanced Powder. So I tried that, and I had some rather positive results!
One caveat. The Finish powder must dissolve quicker than other powders I use, because 4/5 times, if I check the detergent cup for the main wash, most of it is flushed out.
Either way, if I manually add it to the main wash after the prewash on Normal, I've been getting some really nice results with coffee stains being neutralized on almost every mug.
In order to curb the annoyance of adding the powder to the MW, I've been trying the Finish Quantum pacs again, in addition with the Finish powder in the pre-wash, with equally good or better results. The pacs don't pre-flush away through the barely sealing old KA dispenser cover.
The Quantum pacs, with my water, also do not seem to cause excessive foaming in the KA23, like the Cascade Complete and Platinum pacs do.
So I'm not sure what it is. Finish has been working great on my coffee stains.
Maybe they have different ratios of bleaching agents than the others.
On the cups that had lingering stains, I scrubbed them off with BKF, and they have not returned.