Driven by a nucleo board,  the signal from the CCD in the typical drive circuit with a low-noise low drop-out voltage regulator exhibits roughly ±4 mV of noise. This figure is the same for the custom STM32F405 board, however because the opamp has a gain of ca. 2, the S/N-ratio has improved with app. 50%.

Still, the output looks kind of fuzzy, as seen in this figure showing the CCD at close to saturation:

However, because the noise-level is now a little lower, it's become very easy to estimate the CCD's register imbalance, and when correcting the data for it, the same data now looks like this:

I'm not sure I'll get a cleaner picture than this. I'm certainly not in the mood for trying.