You can get creative on the Document sidebox ...
This is restricted to categories that are marked as Document General ...
However ...
You can add to this Product General or any other Product Type ...
Now those categories that are marked Document General show from the Document sidebox ...
The listings etc work the same way but you can separate the categories to be managed by two sideboxes ...
Then change the title of it to whatever is appropriate to your needs for splitting the categories into the two sideboxes ...
NOTE: all categories and the subcategories must be restricted in the same manner for Document General and whatever other Product type you want ... otherwise this will not work properly for you ...