Maybe you could place 3-4 30-60FPS cheapish CMOS cams around the edge of the screen and stagger their frame captures? You'd get different angles (better for detecting eye vectors) and increase the sampling rate.
Why is the IR part of the spectrum better for the cameras? Is it because if I take an image of my eye and look at the IR part of the spectrum is it just easier to see parts of my eye that determine where it's looking?
This is effectively how the FaceID system on your iPhone can work regardless of lighting condition.
That means that even in dark situations, you can have much higher quality imaging , albeit in a limited range.