Hi Hendrik, Brenden,
In the Export you cannot select individual classes to export, only the entire module. Also, have you sealed the module when exported? Also, IsExportable you need to export-import the data stored in this custom class table. Classes itself should be exportable-importable by default (refer to documentation). Are you trying to move data or class? If data, you need to move both module and data selected.
Other option to consider is using Content Staging to migrate those changes. There you can migrate individual classes without the need to re-import the entire module.