Thanks for all the input on this everyone, and a quick update.
As the beers have aged a little more, the taste is changing quite a bit. When I first posted this thread, batches 2 and 3 had only been in the bottle 11 days. Now that it's been 3 weeks, the common background taste has subsided substantially. I can still detect it, but the taste of both have gone from good to very good.
There is a bypass valve on our watersoftener. I think I'll switch that on before I make the next batch and see if it makes a difference.