Here's what I did:
1. Sent a sample of softened water to Ward Labs for analysis. Came back high sodium and basically no other minerals.
2. Sent a sample of water with the bypass pushed in on the softener, Came back with a much richer mineral bill, though iron is high.
I now brew with the hard water exclusively, and it's worlds better. I used to have an acidy, weird aftertaste with the softened water. I guess I could analyze it even further and add distilled to cu some of the iron, but I'm not at that point yet.