Contrary to my earlier assumption, I found a multi-mode I2C chip with 12 ports: the Maxim MAX11312. I've removed the two that I had been using, along with as many references to them as I could find in the Description, and have put in the MAX11312, with a start at the pin connections. I'd be surprised if I don't have to make some changes in that regard (connecting some floating pins, adding some components, changing a connection or two, etc.), but what I have here will do until I can really dive into the data sheet.
I think the board will be much easier to handle with just one I2C address to worry about.