instead of thinking like a human, think like a machine.
work with light and shadow. orient the cameras so they look at the “horizon” of the board plane, where the ring light is. that is the only piece of the picture you need. you also want the ring light to be close to that plane. you’d want to “cut” a plane (from camera to ring light) just a few millimeters above the board plane, so the only thing intersecting the plane is the darts.
a dart, obstructing the ring light, would look dark. if it doesn’t, dial down the exposure until the ring lights are bright but not saturated.
that dark “gap” in the light band is easy to locate. then you can get a ray from the camera through the dart. given three rays (or just two), that’ll be nicely pinned down in 2D.
if that’s not enough, you might have to modify the ring light to be able to light one third of it in turn, the third that is opposite each camera. you’d also need the ability to precisely time the camera exposures, like in a resolution of microseconds or single milliseconds at worst. for that to not look like a strobe light, it’d have to happen quickly, so a whole turn (three thirds) happens in 1/60 s, i.e. each segment lights for 1/180 s or less.
if you use camera modules rather than “webcams”, you might have this ability. the module would have to have a digital input for “exposure”. it might just trigger an exposure, or it might even be able to control the duration of the exposure.
also, maybe solve this electronically rather than visually
