After some back and forth with the creator, we determined that the problem was due to the way I was attempting both the bakeing of the texture, and the import.
It turns out instead of appending the material from the external file and choosing it, I was trying to implement it as part of a pre-existing shaders base color (I AM pretty new to all this).
The correct procedure is to Append the CW eye material, and then choose it as the new material itself:

The baking issue was due to me not having understood that the object one bakes to has to be superposed with the object itself !

Thank you to Christop Werner for his help in figuring out how to properly use his awesome material !