Up until now, most photocurrent I was getting from the phototransistor was because of the light bouncing off the lenses of the eyepiece, and probably a fair amount from reflection of the eye's sclera (the white stuff) and skin. I think these "parasitic" reflections completely obscured the signal from the red eye (obviously they are dependent on the position of the sensor). And I have a few thoughts on how to overcome this.

First. The red-eye return light is focused, while the parasitic light is not. So, if I place another photodiode close to the main one, it will receive a nearly identical parasitic light, but not the red-eye return signal. So, subtract the signals from those two photodiodes - and the signal from parasitic light is gone. That's the plan. But I have to switch to photodiodes (not phototransistors) at this point, because sensitivities must match.

Second, I should limit the cone from where the red-eye light is collected to as much close to only the eye's pupil as possible. That's not easy, but since the eye's pupil must be in quite specific position in order to see through the microscope, that is doable.