Finally broke down and bought a Berkey water filter.
A Travel Berkey.
The Brita thing just wasn't working anymore. Not only are the filters expensive and only last a couple of months each, but our refrigerator in our new apartment is significantly smaller than the previous fridges, and there is just no room for those big Brita pitchers.
The water coming out of the new fridge is just a little suspicious, so another solution was needed. Berkey claims that the filters that came with the Travel Berkey model will last for about 3,000 gallons of water each. It uses two filters at the same time.
So, based on our estimated consumption of about 1/2 gallon of water per day while we are at home, I figure that these filters will last for over 15 years. And that translates to about $0.01 per 16 oz of water. I think I can handle that!