As you know, FBX is a proprietary format, and the Blender importer/exporter is limited in what it can export to an FBX file.
You can read the Python source code of the FBX exporter on github. In particular, the function that exports materials is at line 1284: fbx_data_material_elements
.
Many of Blender's material properties are ignore, or reduced to a few values. As Jachym Michal said in a comment the Opaque mode property is one of those that are ignored.
You have one slim hope of being able to do what you want, hinted at by line 1357 - 1358: You can add a custom property that encodes the Opaque mode, driven by a driver.
But for this to work you would have to write a wrapper for the importer that rebuilt the custom property and added the driver; and, of course, only your modified importer would understand it.
I'm afraid you're out of luck.