Today I gave up trying to fix the first board I populated, and built a second one. The good news is that it's working as expected.
With 27.5 uW incident on each quadrant, I measure about 4.5 V, which is in line with the transimpedance of 200k, a responsivity of about 0.4 A/W and ad additional gain of two coming from the differential driver.
I also measured the noise with a SR785 (it wasn't connected to a GPIB interface and I couldn't find any, so all I have are the following numbers and the attached screenshots).
At low frequency we are dominated by 60 Hz harmonics (probably coming from the laser). At high frequency there are some large peaks of unknown origin. I can't acquire digitally the signals to compute the difference, so I don't know if the noise we see is, for example, laser intensity noise. As soon as the cymac is up and running, I'll run some more tests.