You’ve gotten some good information so far; here are my thoughts.
The eye is roundish and highly reflective due to the constant bathing in fluid. Normally catch lights are desired, except in your circumstance. Unfortunately being a highly reflective round surface means the family of angles is huge and trying to eliminate them would be akin to herding cats; darn near impossible. A source of light anywhere in front or to the sides is most likely going to cause a spectral highlight.
My thought is that if you can’t beat it, control it. The normal practice is to diffuse and make the source larger to minimize shadows and harsh lighting. But in this case I don’t see harsh lighting as being too much of an issue; and your diffused light source would need to be huge to cover the family of angles. Making the light source as small as possible helps you control the catch light size as well as the placement.
This is the setup I devised. I used index cards with the face blackened out with a small hole cut in it. I used the Nikon supplied diffuser covered with a black microfiber cloth to minimize any light spill and taped the index card to the front. I then placed it as on axis to the lens as I could. I would have liked to get the flash a little further forward, but I don’t have a boom for my light stand and didn’t feel like resetting up with the tripod legs entangled (I was doing this by myself as the subject and photographer which was a pain!!).
And this was the result. Although there still is a catch light, it is in the pupil area which is not detrimental to your task I believe. If you are the photographer with a separate subject and want to take the time, you could probably even use a small GOBO to block that light source and eliminate the spectral highlight without impacting the overall illumination.