I think if there was an easy solution people would have cracked the formula by now. The biggest problem with depicting anime/cartoons in 3D is that what works on paper doesn’t translate 100% into 3D space. What I mean is that while the proportions of anatomy are kept consistent between angles their depiction isn’t and “morphs” depending on our perspective, which works and is much easier to do in 2D than it is in 3D.
What anime would look like without morphing:
For best results, my best advice here would be to look at the most successful depictions of 3D anime characters and try to copy what they’ve done. Think Genshin Impact, Xenoblade Chronicles 3 or recent works from studio MAPPA such as Chainsaw Man or 2nd season of Zombieland Saga. Though I think most of them use some form of 2D eyes.
That said, while I’m not sure what the model is like in this course, I don’t see what’s stopping you from just using a couple of spheres for the eyes and shaping the eye sockets around it. This is what I did for my model, though I was going for a more “Disney” style:
If you cannot fit a full sphere inside the skull you can go for partial ones, cut off the excess and only keep a small amount to show the roundness of the eye. This is what I did for this chibi version:
You can leave a little more excess if you want to rotate the actual geometry of the eye, but in my case, I settled for “animating” the texture instead.
I think there is some potential to do something smart here with the Decal Machine plug-in but I’m not willing to experiment right now. Decal machine lets you bake 3D geometry onto a flat plane that then shifts in what it displays based on the angle you look from. It’s most commonly used to compliment hard surface modelling or quickly stamp extra details but I can somehow see it potentially working quite well for anime eyes.