It took a while, but I eventually noticed that the modulator's output signal zero-crosses had little in common with the sine-cosine zero-crosses. If only there was a way to shift them a little bit...

Oh, right. Analog, sine waves, phase shift! Only 45 degrees is needed, this shouldn't be too difficult. Many simulations later, I changed a few resistor values and added two capacitors and now the zero-crosses are all lined up.

The updated schematic/KiCAD project has been uploaded.