I have noticed that if you use Google Backup & Sync instead of Drive File Stream you can back this folder up using Time Machine (no compulsory greyed out option). Note: I have only tested that this works in the ~/CustomFolderName/ folder but I assume it works for any other folder than the default.
You can do this inside the DFS 'preferences' menu. The only way to force Time Machine to include this cache in the backup is by moving the cache to another location. However, please note that Time Machine also excludes this cache by default. Unfortunately, this cache is not human readable but it can be mounted by Drive File Stream into the human readable offline file structure that you are familiar with seeing (in fact this is exactly what DFS does once you log in). Mac: ~/Library/Application Support/Google/DriveFS When you make your files available for offline access, Drive File Stream (DFS) stores a cached version of this data in the following locations: I am facing the same issue and have found a clean (recommended) and a messy solution.
