Instead of converting to Shapefile and referencing that, do it the other way around:
Load your CAD file, this will be inserted as a group layer.
Select on of the actual layers (eg Multipatch) to activate the CAD-Data ribbon
You can now georeference the CAD file like you would do with imagery. When you're done, you can export to the format of your choice.
https://pro.arcgis.com/en/pro-app/3.0/help/data/cad/georeferencing-cad-data.htm
Have a great day!
Johannes