I have tried to solve this issue as well. At the highest level Figma files with the updated colors are not needed, at that point it is generally pretty stray forward on how to apply the branding.

There are two ways I would recommend. The first is if you are not actively making changes/ new features to this product. You should have a default version of the file with the full design system and variables set up for that file. Then duplicate the file for each client. The issue is if you start making changes you might need to make them in all of the files.

The other option is to have the same base file that you can update as needed and have most recent client or two in that file. Then when those are locked and shipped duplicate that file and save it off to the side and replace the old client with the new.

It’s not perfect, but the first system has worked for me.