It turns out that correcting the mode of the STA120 was not enough. The audio sounded way distorted. The music was identifiable, but it was very, very fuzzy. I tried playing with the !DEM/SCLK pin, but that didn't change anything. The steady state of the de-emphasis output from the STA120 was off (high) anyway, so on a whim I hooked !DEM/SCLK up to the SCLK pin of the STA120 and that made the audio perfect.

So 3 total changes: adding a series inductor to the supply pin of the optical module, fixing the mode pins on the STA120, and hooking the SCLK pin between the two.

I'm going to order one more set of boards from OSHPark and call it a day.