It is very sad what Autodesk is not opening the Assembly code system to be able to use any other classification systems. Instead, they created additional tools like BIM Interpobility Tools for classification. And this system is not linked with the IFC exporter. It would be great if they just delete Assembly code and replace it with a more flexible tool like the Interpobility tool. And of course, they should invest more time and for on Revit and IFC integration.
At the moment you can modify IFC by hand using Notepad++ you should change information in this location:
#283= IFCCLASSIFICATION('<Link to Clasification source page>','<version number>','<Edition Date>','<Clasification name>','<Clasification Description>','<Location>','<??>');
#286= IFCCLASSIFICATIONREFERENCE('<Clasification oficila page link>','<Clasification code>','<Clasification Name>',#283,'<Clasification Description>','<??>');
#291= IFCRELASSOCIATESCLASSIFICATION('3vD1HL2uj2UAQQzOHfCkfz',#48,'<Clasification Name>','',(#272,#310),#286);
Below present how it should look then you export to IFC.
Before modification:
After modification: