I built the next prototype. While the boards were being fabbed, I found an error in the schematic - for the left channel, the + and - differential outputs from the DAC were reversed. I cut the traces on the board and bodged them back.
But the audio of the prototype sounded bad. After spending the afternoon trying to figure it out, it turns out that the problem was that I gave the final amplifier a unipolar power supply (that is, the V- pin of the amp was grounded.
Lifting that pin and temporarily hacking in a 5 volt wall wart with - to V- and + to ground gave correct audio (albeit with a nasty 60 Hz hum).
The problem now is that everybody I've found on the Internet who has made the schematic of their DAC public has a bipolar power supply.
Well, you can make -5 volts from +5 with a charge pump. The MAX1721 has a switching frequency of 125 kHz, so ostensibly above anything audible. Hacking in a MAX1720 on the prototype did make it work. But if it's possible, rebiasing the output amp to operate on a 2.5 volt offset, reducing the gain and adding a DC block on the output would probably be better for the THN spec of the result.
The next prototype board (boards?) will test one or both of these options.